diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzqgwm" "b/data_all_eng_slimpj/shuffled/split2/finalzzqgwm"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzqgwm"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n Stationarity is crucial in analyzing random sequences because statistical inference usually requires a probabilistic mechanism constant in, at least, a segment of observations.  Therefore, it is important to detect whether changes occur in a sequence of observations prior to statistical inference.  Such change-point problems arise from many applications: abrupt events in video surveillance \\citep{mayer2015change};   deterioration of product quality in quality control \\citep{lai1995sequential}; credit card fraud in finance \\citep{bolton2002statistical}; alteration of genetic regions in cancer research \\citep{erdman2008fast} and so on.\n\nThere is a large and rapidly growing literature on change-point detection \\citep{aminikhanghahi2016survey,niu2016multiple,sharma2016trend,fryzlewicz2014wild}. Many methods rely on the assumed parametric models to detect special change types  such as location, scale or presumed distribution family. \\citet{page1954continuous} introduced a method by examining the ratio of log-likelihood functions. \\citet{lavielle2006detection} detected change-points by maximizing a log-likelihood function. \\citet{yau2016inference} proposed a likelihood ratio scan method for piecewise stationary autoregressive time series. Some Bayesian change-point detection methods assume that the observations are normally distributed, and calculate the probability of change-point at each point \\citep{barry1993bayesian,Zhang1995changepoint,wang2015bayesian,maheu2018efficient} to name a few. Since parametric methods potentially suffer from model misspecification, other methods are developed to detect general distributional changes with more relaxed assumptions.\n\\citet{kawahara2012sequential} provided an algorithm which relied heavily on estimating the ratio of probability densities. \\citet{lung2015homogeneity} identified change-points via the well-known Wilcoxon rank statistic. \\citet{matteson2014nonparametric} proposed a nonparametric method using the concept of energy distance for independent observations. There are also some binary segmentation methods statistics  \\citep{fryzlewicz2014wild,cho2015multiple,eichinger2018mosum}. Two advantages of the binary segmentation procedures are their simplicity and computational efficiency, but their false discovery rates may be hard to control because they are `greedy' procedures. \\citet{zou2014nonparametric} introduced a nonparametric empirical likelihood approach to detecting multiple change-points in independent sequences, and estimated the locations of the change-points by using the dynamic programming algorithm and the intrinsic order structure of the likelihood function.\n\nAutomatically detecting the number of change-points is also important.\nSome methods are developed to detect only a single change-point \\citep{ryabko2008hypotheses}, while some methods require a known number of change-points but unknown locations \\citep{hawkins2001fitting,lung2015homogeneity}. In real data analysis, however, we usually do not know the number of change-points.\n\nWith increasing richness of data types,  non-Euclidean data, such as shape data, functional data, and spatial data, commonly arise from applications. For example, one of the problems of interest to us is the changes in the monsoon direction as defined by circle, a simple Riemannian manifold. Methods developed in Hilbert spaces are not effective for this type of problems as our analysis of the data from Yunnan-Guizhou Plateau ($105^\\circ$E, $27^\\circ$N) collected from 2015\/06\/01 to 2015\/10\/30 illustrates below.  To the best of our knowledge, few methods exist to detect change-points in a non-Euclidean sequence.  \\citet{chen2015graph} and \\citet{chu2019asymptotic} proposed a series of graph-based nonparametric approaches that could be applied to non-Euclidean data with arbitrary dimension. However, their proposed  methods apply to \\emph{iid} observations only and are restricted to one or two change-points. Therefore, it remains to be an open and challenging problem to develop methods to detect arbitrarily distributional changes for non-Euclidean sequences, including the change-point locations and the number of the change-points.\n\nTo address this challenge,  we introduce a novel concept of Ball detection function via Ball divergence \\citep{pan2018ball}. Ball divergence  is a recently developed measure of divergence between two probabilities in separable Banach spaces. The Ball divergence is zero if and only if the two probability measures are identical. Since its sample statistic is constructed by metric ranks, the test procedure for an identical distribution is robust to heavy-tailed data or outliers, consistent against alternative hypothesis, and applicable to imbalanced data. Therefore, the empirical ball divergence is an ideal statistic to test whether or not a change has occurred. Unfortunately, it does not inform us where the change occurs, because in theory the probability measures before and after any time point are always different if there exists a change point in the sequence. Therefore it is imperative for us to observe how the probability measures before and after any time vary with time and then develop a proper criterion to detect the change-point location. We introduce a Ball detection function as an effective choice which reaches its maximum at the change point if a sequence has only one change point. We further develop a hierarchical algorithm to detect multiple change-points using the statistic based on the Ball detection function. The advantages of our procedure are threefold: our procedure can  estimate the number of change-points and detect their locations;  our procedure can detect any types of change-points; and both uniquely and importantly, our procedure can handle complex stochastic sequences, for example, non-Euclidean sequences.\n\nThe rest of this article is organized as follows. In Section 2, we review the notion of Ball divergence,  and then introduce a novel change-point detection function, i.e., a Ball detection function based on Ball divergence with a scale parameter for weakly dependent sequences. We further establish its asymptotic properties. We show how to use the Ball detection function to detect change-points and  establish the consistent properties of our method in Section 3.  In Section 4, we compare the performance of our method with some existing methods in various simulation settings. In section 5, two real data analyses demonstrate the utility  of our proposed method. We make some concluding remarks in Section 6. All technical details are deferred to Appendix.\n\n\\section{Change-point Detection in Dependent Sequences}\n\n\\subsection{Review of Ball Divergence}\n\n\nBall divergence (BD, \\citet{pan2018ball})  is a measure of the difference between two probabilities in a separable Banach space $(\\mathrm{A},||\\cdot||)$, with the norm $||\\cdot||$. $\\forall~u,v \\in \\mathrm{A}$, the distance  between $u$ and $v$ deduced from the norm is $\\rho(u,v)=|| u-v ||$. Denote by $\\bar{B}(u,r)=\\{x|\\rho(x,u)\\leq r\\}$  a closed ball. Let $\\mathcal{B}$ be the smallest $\\sigma$-algebra in $\\mathrm{A}$ that contains all closed (or open) subsets of $\\mathrm{A}$. Let $\\mu$ and $\\nu$ be two probabilities on $\\mathcal{B}$. Ball divergence \\citep{pan2018ball} is defined as follows.\n\n\\begin{definition}\\label{df_bdo}\n\tThe Ball divergence of  two Borel probabilities $\\mu$ and $\\nu$ in $\\mathrm{A}$ is defined as an integral of the square of the measure difference between $\\mu$ and $\\nu$ over  arbitrary closed balls,\n\t\\begin{equation*\n\tD(\\mu,\\nu)=\\iint_{\\mathrm{A}\\times \\mathrm{A}} [\\mu-\\nu]^2 (\\bar{B}(u,\\rho(u,v)))(\\mu(du) \\mu(dv)+ \\nu(du)\\nu(dv)).\n\t\\end{equation*}\n\\end{definition}\n\nLet $S_\\mu$ and $S_\\nu$ be the support sets of $\\mu$ and $\\nu$ respectively. The BD has the following important property \\citep{pan2018ball}:\n\n\\begin{theorem}\\label{quantity} Given two Borel probabilities $\\mu$ and $\\nu$ in a  finite dimensional Banach space $\\mathrm{A}$, then $D(\\mu,\\nu)\\geq 0$  where the equality holds if and only if $\\mu=\\nu$.  It can be extended to separable Banach spaces if $S_\\mu=\\mathrm{A}$ or $S_\\nu=\\mathrm{A}$.\n\\end{theorem}\n\n\\subsection{Ball Divergence with a Scale Parameter}\n\nThe Ball divergence introduced above cannot detect the locations of change-points accurately enough while comparing the distributions of the sequences before and after the change-points. We need to introduce a  Ball divergence associated with a scale parameter $\\alpha$ as follows.\n\n\\begin{definition}\\label{df_bd} A Ball divergence of two Borel measures $\\mu$ and $\\nu$ in $\\mathrm{A}$ is defined as\n\t\\begin{align}\\label{formula_alphaBD}\n\tD_{\\alpha}(\\mu,\\nu)\n\t=\\iint_{\\mathrm{A}\\times \\mathrm{A}} [\\mu-\\nu]^{2}(\\bar{B}(u,\\rho(u,v))\\omega_{\\alpha} (du)\\omega_{\\alpha} (dv),\n\t\\end{align}\n\twhere $\\omega_{\\alpha}=\\alpha \\mu+(1-\\alpha) \\nu$ is the mixture distribution measure with the scale parameter $\\alpha\\in[0,1]$.\n\\end{definition}\n\n$D_{\\alpha}(\\mu,\\nu)$ also has the equivalence property below, which is critical to the comparison of the distributions of any two sequences.\n\\begin{theorem}\\label{BDifandonlyif}\n\tGiven two Borel probabilities $\\mu,\\nu$ in a finite dimensional Banach space $\\mathrm{A}$, then $D_{\\alpha}(\\mu,\\nu)\\geq 0$ where the equality holds if and only if $\\mu=\\nu$. It also holds on separable Banach spaces if $S_\\mu=\\mathrm{A}$ or $S_\\nu=\\mathrm{A}$.\n\\end{theorem}\n\nTheorem \\ref{BDifandonlyif} assures that for any $\\alpha\\in [0,1],$ $D_{\\alpha}(\\mu,\\nu)$ possesses the most important property as $D(\\mu,\\nu)$ in terms of\ntesting the distributional difference between two sequences.  Importantly, with the introduction of $\\alpha$, we can consistently estimate the locations of the change-points. Here, we highlight the relationship and difference between $D_{\\alpha}(\\mu,\\nu)$ and $D(\\mu,\\nu).$\n\nWhen $\\alpha=1$, $D_{1}(\\mu,\\nu)$ is the measure difference over the balls whose centers and the endpoints of the radius following measure $\\mu.$ When $\\alpha=0$, $D_{0}(\\mu,\\nu)$ is the measure difference over the balls whose centers and the endpoints of the radius following the measure $\\nu$. Moreover,\n\\begin{equation*}\nD(\\mu,\\nu)=D_{0}(\\mu,\\nu)+D_{1}(\\mu,\\nu).\n\\end{equation*}\nFor $\\alpha\\in(0,1)$, $D_{\\alpha}(\\mu,\\nu)$ is the mean of the measure differences from two samples over the balls whose centers and endpoints of the radius following four possible pairs of measures:$(\\mu,\\mu)$, $(\\mu,\\nu)$,$(\\nu,\\mu)$, and $(\\nu,\\nu)$ where the ratio of two measures is $\\alpha:1-\\alpha$.\n\n\nBall divergence with a  scale parameter can be defined in the general metric space, following the Generalized Banach-Mazur theorem \\citep{kleiber1969a} as stated in the Supplementary material.\n\n\\subsection{Ball Detection Function}\nNow, we introduce a Ball detection function which is maximized at the change point if there exists one, and hence can be used to\ndetermine the location of the change point. For clarity, let us consider a conceptual sequence with a change point $\\alpha \\in (0,1)$, and the probability measures before and after $\\alpha$ are $\\mu$ and $\\nu$, respectively.  Denote the indicator function by $I(\\cdot)$. For  a \"time\" $\\beta \\in (0,1)$, define \n\\begin{equation*}\n  h_{\\alpha}(\\beta)=\\frac{\\alpha}{\\beta}I(\\beta\\geq\\alpha)+\\frac{1-\\alpha}{1-\\beta}I(\\beta<\\alpha).\n\\end{equation*}\nWithout loss of generality, suppose that $\\beta>\\alpha$, the probability measures before and after $\\beta$ are $\\frac{\\alpha}{\\beta}\\mu+(1-\\frac{\\alpha}{\\beta})\\nu$ and $\\nu$. By the definition of Ball divergence (\\ref{formula_alphaBD}), we have\n\\begin{equation*}\nD_{\\beta}(\\frac{\\alpha}{\\beta}\\mu+(1-\\frac{\\alpha}{\\beta})\\nu,\\nu)=(\\frac{\\alpha}{\\beta})^2D_{\\alpha}(\\mu,\\nu).\n\\end{equation*}\nTherefore, in general,\n\\begin{equation}\\label{relation alpha-beta}\nD_{\\beta}(h_{\\alpha}(\\beta)\\mu+(1-h_{\\alpha}(\\beta)\n)\\nu,\\nu)=h^2_{\\alpha}(\\beta)D_{\\alpha}(\\mu,\\nu).\n\\end{equation}\nThe maximum of $h_{\\alpha}(\\beta)$ is attained when $\\beta=\\alpha$ if there exists a change-point $\\alpha.$ In this case, we can find the change point by maximizing the ball divergence in equation (\\ref{relation alpha-beta}). But we still need to test whether a change point has occurred or not. Next, we introduce a Ball detection function to simultaneously test the existence of a change-point and determine its location:\n\\begin{eqnarray*}\\label{relation Ball-detection}\nV(\\beta;\\mu,\\nu) &=&\\beta(1-\\beta)D_{\\beta}(h_{\\alpha}(\\beta)\\mu+(1-h_{\\alpha}(\\beta)\n)\\nu,\\nu)\\\\&=&\\beta(1-\\beta)h^2_{\\alpha}(\\beta)D_{\\alpha}(\\mu,\\nu).\n\\end{eqnarray*}\nNote that the maximum of $\\beta(1-\\beta)h_{\\alpha}(\\beta)$ is also attained when $\\beta=\\alpha$, allowing us to find the change point by maximizing $V(\\beta;\\mu,\\nu)$. In next subsection, we shall discuss how this function is used to construct a test for a change-point test statistic.\n\n\n\\subsection{Ball Detection Function in Sample}\nSuppose that a sequence of observations $\\{Z_i\\}_{1 \\leq i \\leq T}$ is comprised of  two multivariate stationary sequences $\\{Z_i\\}_{1 \\leq i \\leq M}$ with the probability measure $\\mu_1$ and $\\{Z_i\\}_{M+1\\leq i \\leq T}$ with $\\mu_2$, where  both  $\\mu_1$ and $\\mu_2$ are unknown. We estimate $D_{\\alpha}(\\mu_1,\\mu_2)$ with $\\alpha=M\/T$ based on $\\{Z_i\\}_{1 \\leq i \\leq T}$.  Let $c(x,y;z)=I(z\\in \\bar{B}(x,\\rho (x,y)))$, which identifies whether the point $z$  falls into the closed ball $\\bar{B}(x,\\rho (x,y))$ with $x$ as the center and $\\rho (x,y)$ as the radius, and $e(x,y,z_1,z_2)=c(x,y;z_1)c(x,y;z_2)$, which determines whether two points $z_1$ and $z_2$ fall into the ball $\\bar{B}(x,\\rho (x,y))$ together.   Let $N=T-M$, $C_{ij}^{1}=\\frac{1}{M}\\sum_{u=1}^Mc(Z_i,Z_j;Z_{u}),\nC_{ij}^{2}=\\frac{1}{N}\\sum_{v=M+1}^Tc(Z_i,Z_j;Z_{v}).$\nA consistent estimator of the Ball divergence of $\\mu_1$ and $\\mu_2$  with the scale parameter $\\alpha$ is\n\\begin{equation*}\nD_{M,N}=\\frac{1}{T^2}\\sum\\limits_{i,j=1}^{T}(C_{ij}^{1}-C_{ij}^{2})^2,\n\\end{equation*}\nas summarized in Theorem \\ref{slln}.\n\nWe also prove that $\\frac{MN}{T}D_{M,N}$ has a limiting distribution under the null hypothesis in Theorem \\ref{h0}. For this reason, we choose\n\t\\begin{equation*}\n\tV(M,T)=\\frac{MN}{T}D_{M,N}\n\t\\end{equation*}\nas the statistic to detect change-points.\n\nTo investigate the asymptotic properties of $V(M, T)$, we introduce two concepts of the random sequence: absolutely regular and ergodic stationary sequence.\n\nGiven the probability space $(\\Omega,\\mathscr{F},P)$ and two sub-$\\sigma$-fields $\\mathscr{A}$ and $\\mathscr{B}$ of $\\mathscr{F}$, let\n$$\n\\beta(\\mathscr{A},\\mathscr{B})=\\sup\\sum_{i=1}^m\\sum_{j=1}^n|P(A_{i}\\bigcap B_{j})-P(A_{i})P(B_{j}))|,\n$$\nwhere the supreme is taken over all partitions of $\\Omega$ into sets $A_1,\\ldots,A_m\\in\\mathscr{A}$, all partitions of $\\Omega$ into sets $B_1,\\ldots,B_n\\in\\mathscr{B}$ and all $m,n\\geq1$. A stochastic sequence $\\{Z_i\\}_{i\\in\\mathds{Z}}$ is called absolutely regular ( \\citep{dehling2012asymptotic}, also called weakly Bernoulli \\citep{aaronson1996strong}), if\n$$\n\\beta(l)=\\sup\\limits_{n}\\beta(\\mathscr{F}_{0}^n,\\mathscr{F}_{n+l}^\\infty)\\rightarrow 0,\n$$\nas $l\\rightarrow\\infty$. Here the $\\mathscr{F}_i^j$ denotes the $\\sigma$-field generated by the random variables $Z_i,\\ldots,Z_j$. In this paper, we suppose that $\\beta(l)=O(l^{-1-r})$ for any $r>0$. The concept of absolutely regular sequence is wide enough to cover all relevant examples from statistics except for long memory sequences.\n\nRecall that an ergodic, stationary sequence (ESS) \\citep{aaronson1996strong} is a random sequence $\\{Z_i\\}_{1\\leq i\\leq T}$ of form $Z_{i}=f(G^i)$ where $G^i$ is an ergodic, probability-preserving transformation in the probability space $(\\Omega,\\mathscr{F},P)$, and $f$ is a measurable function.\nIn essence, an ESS implies that the random sequence will not change its statistical properties with time (stationarity) and that its statistical properties  can be deduced from a single, sufficiently long sample of the sequence (ergodicity).\n\nWe have the following theorem for an absolutely regular sequence comprised of two ergodic stationary sequences:\n\n\\begin{theorem}\\label{slln} Suppose that $\\{Z_i\\}_{1 \\leq i \\leq T}$ is an absolutely regular sequence, $\\{Z_i\\}_{1 \\leq i \\leq M}$ and $\\{Z_i\\}_{M+1\\leq i \\leq T}$ are both ergodic stationary with marginal probability measure $\\mu_1,\\mu_2$ respectively. When $M,T\\rightarrow \\infty$, $M\/T\\rightarrow\\alpha_1$ for some $\\alpha_1\\in[0,1]$,  then\n\t$$\\frac{V(M,T)}{T}\\xrightarrow[M,T\\rightarrow\\infty]{a.s.}V(\\alpha_1;\\mu_1,\\mu_2).$$\n\\end{theorem}\n\nTheorem \\ref{slln} means that $\\frac{V(M,T)}{T}$ converges to Ball detection function $V(\\alpha_1;\\mu_1,\\mu_2)$ almost surely. We further investigate the asymptotic distribution of $V(M,T)$. Under the null hypothesis, the Ball detection function in sample is the sum of four degenerate V-statistics. As in \\cite{pan2018ball}, we denote $Q(x,y;x',y')$ as the second component in the H-decomposition of $V(M,T)$. Then we have the spectral decomposition:\n$$Q(x,y;x',y')=\\sum_{k=1}^{\\infty}\\lambda_kf_k(x,y)f_k(x',y'),$$\nwhere $\\lambda_k$ and $f_k$ are the eigenvalues and eigenfunctions of $Q(x,y;x',y')$. Let $\\{Z_i'\\}_{1\\leq i\\leq T}$ be an independent copy of $\\{Z_i\\}_{1\\leq i\\leq T}$. For $k\\in\\{1,2,\\ldots\\},$    $N_{1k},N_{2k}$ are assumed to be \\emph{iid} $N(0,1)$, and let\n\\begin{align*}\na_k^2(\\alpha_1)=(1-\\alpha_1)E_{Z_1}[E_{Z_1'}f_k(Z_1,Z_1')]^2,\\quad\nb_k^2(\\alpha_1)=\\alpha_1E_{Z_1'}[E_{Z_1}f_k(Z_1,Z_1')]^2,\\\\\nc_k^2(\\alpha_1)=a_k^2(\\alpha_1)+2(1-\\alpha_1)(\\sum_{j=1}^{\\infty}E_{Z_1,Z_{1+j}}[E_{Z_1'}f_k(Z_1,Z_1')E_{Z_1'}f_k(Z_{1+j},Z_1')]),\\\\\nd_k^2(\\alpha_1)=b_k^2(\\alpha_1)+2\\alpha_1(\\sum_{j=1}^{\\infty}E_{Z_1',Z_{1+j}'}[E_{Z_1}f_k(Z_1,Z_1')E_{Z_1}f_k(Z_1,Z_{1+j}')]),\n\\end{align*}\n$$\\theta=E[E(c(Z_1,Z_2,Z_i)(1-c(Z_1,Z_2,Z_{j}))|Z_1,Z_2)].\n$$\n\\begin{theorem}\\label{h0}\n\tUnder null hypothesis $H_{0}: \\mu_1=\\mu_2$, $\\{Z_i\\}_{1\\leq i\\leq T}$ is a stationary absolutely regular sequence with coefficients satisfying $\\beta(l)=O(l^{-1-r})$ for $r>0$, if  $M,T\\rightarrow \\infty$, $M\/T\\rightarrow\\alpha_1$ for some $\\alpha_1\\in[0,1]$, we have \n$$V(M,T)\\xrightarrow[M,T\\rightarrow\\infty]{d}\\sum_{k=1}^{\\infty}\\lambda_{k}[(c_{k}(\\alpha_1)N_{1k}+d_{k}(\\alpha_1)N_{2k})^{2}-(a_{k}^{2}(\\alpha_1)+b_{k}^{2}(\\alpha_1))]+\\theta. $$\n\\end{theorem}\n\nUnder the alternative hypothesis, the Ball detection function in sample  is asymptotically normal because it is a sum of non-degenerate V-statistics. Let $g^{(1,0)}(Z_\\mu)$ and $g^{(0,1)}(Z_{\\nu})$ be the first component in H-decomposition of $V(M,T)$ and\n$$\\delta_{1,0}^2=Var(g^{(1,0)}(Z_u))+2\\sum_{i=1}^\\infty Cov(g^{(1,0)}(Z_u),g^{(1,0)}(Z_{u+i})),$$\n$$\\delta_{0,1}^2=Var(g^{(0,1)}(Z_v))+2\\sum_{i=1}^\\infty Cov(g^{(0,1)}(Z_v),g^{(0,1)}(Z_{v+i})).$$\n We can obtain the asymptotic distribution under the alternative hypothesis.\n\\begin{theorem}\\label{h1}\n\t$\\{Z_i\\}_{1\\leq i\\leq T}$ is a absolutely regular sequence with coefficients satisfying $\\beta(l)=O(l^{-1-r})$ for $r>0$. Under $H_{1}: \\mu_1\\neq  \\mu_2$, if $M,T\\rightarrow \\infty$, and  $M\/T\\rightarrow\\alpha_1$ for some $\\alpha_1\\in[0,1]$, then we have\n\t$$\\sqrt{\\frac{T}{MN}}(V(M,T)-TV(\\alpha_1;\\mu_1,\\mu_2))\\xrightarrow[M,T\\rightarrow\\infty]{d} N(0,(1-\\alpha_1)\\delta_{1,0}^2+\\alpha_1\\delta_{0,1}^2).$$\n\\end{theorem}\n\n\nWe show that the Ball detection function in sample is consistent against general alternatives. Our new detection function can handle the problem of imbalanced sample sizes. As shown in the following theorem, the asymptotic power of the test does not go to zero even if $\\eta=\\frac{M}{N}$ goes to $0$ or $\\infty$.\n\n\\begin{theorem}\\label{againstall}\n\tThe test based on $V(M,T)\/T$ is consistent against any general alternative $H_1$. More specifically,\n\t$$\n\t\\lim\\limits_{(M,T)\\rightarrow\\infty}Var_{H_1}(V(M,T)\/T)=0,\n\t$$\n\tand\n\t$$\n\t\\Lambda:=\\liminf\\limits_{(M,T)\\rightarrow\\infty}(E_{H_1}V(M,T)-E_{H_0}V(M,T))\/T>0.\n\t$$\n\\end{theorem}\n\n\n\\section{Detection of change-points}\n\\subsection{Hierarchical Algorithm}\n\nNext, we use the Ball detection function in sample to detect change-points in a sequence.\nFor simplicity, suppose that the sequence $\\{Z_i\\}_{1\\leq i\\leq T}$ contains at most one change-point.\nThe possible change-point location is then estimated by maximizing the detection function:\n\\begin{equation}\\label{singleLoc}\n\\hat{M}_1=\\argmax_{M}V(M,T).\n\\end{equation}\nWe use the bootstrap method to estimate the probability that $V(\\hat{M}_1,T)$  exceeds a threshold.\nIf the estimated probability is high enough, $\\hat{M}_1$ is the estimated change-point. Otherwise, we proceed as if there does not exist any change-point in the sequence.\n\nIt is more complicated if the sequence has multiple change-points. In this case, we estimate the first change-point by\n\\begin{equation}\\label{singleLocLLL}\n(\\hat{M}_1,\\hat{L}_1)=\\argmax_{0<M_1<L_1\\leq T}V(M_1,L_1).\n\\end{equation}\nFrom (\\ref{singleLocLLL}), we can see that the introduction of $L_1$ here is to alleviate a weakness of bisection algorithm  \\citep{matteson2014nonparametric}. Because in each segment, there may exist multiple change-points. If we do not introduce $L_1$,  the value of  $V(\\hat{M}_1,T)$ may be lower than $V(\\hat{M}_1,\\hat{L}_1)$.\n\nSuppose that $k-1$ change-points have been estimated at locations $0<\\hat{T}_1<\\cdots<\\hat{T}_{k-1}<T$, and $\\hat{T}_0=0$, $\\hat{T}_k=T$. Those change-points partition the sequence into $k$ segments $\\mathbf{Z}(\\hat{T}_1\/\\hat{T}_0),\\ldots,\\mathbf{Z}(\\hat{T}_k\/\\hat{T}_{k-1})$.\nIn segment $i$, let $C_{ij}^1=\\frac{1}{M_i-\\hat{T}_{i-1}}\\sum_{u=\\hat{T}_{i-1}}^{M_i}c(Z_i,Z_j,Z_u),$ $C_{ij}^2=\\frac{1}{L_i-M_{i}}\\sum_{v=M_{i}}^{L_i}c(Z_i,Z_j,Z_u).$\nThe Ball detection function in sample of segment $i$ is denoted as\n$$\nV_i(M_i,L_i)=\\frac{(M_i-\\hat{T}_{i-1})(L_i-M_i)}{(L_i-\\hat{T}_{i-1})^3}\\sum_{i=\\hat{T}_{I-1}}^{L_i}(C_{ij}^1-C_{ij}^1)^2.\n$$\nNow let\n\\begin{equation}\\label{kappaformula}\n(\\hat{M}_{\\hat{i}},\\hat{L}_{\\hat{i}})=\\argmax\\limits_{1\\leq i\\leq k-1,\\hat{T}_{i-1}<M_i<L_i \\leq \\hat{T}_i}V_i(M_i,L_i).\n\\end{equation}\nThen  $\\hat{M}_{\\hat{i}}$ is the $k$-th possible change-point located within segment $\\mathbf{Z}(\\hat{T}_{\\hat{i}}\/\\hat{T}_{\\hat{i}-1}).$\nThis hierarchical algorithm for estimating multiple change-points is outlined below.\n\\begin{algorithm}\\label{algorithm}\n\t\\caption{Multiple change-points Algorithm}\n\t\\begin{algorithmic}\n\t\t\\STATE Let the minimum segment size $min=m$, the change-points set $\\mathbf{T}=\\{0,T\\}$.\n\t\t\\STATE Suppose that $k-1$ change-points have been estimated. This decomposes the observations into $k$ segments.\n\t\t\\FOR {each $i\\in \\{1,\\ldots,k-1\\}$}\n\t\n\t\t\\IF {the $i-$th segment $\\mathbf{Z}(\\hat{T}_i\/\\hat{T}_{i-1})$ is a new segment,}\n\t\t\\STATE $best_i=0;$\n\t\t\\FOR{$M_i=\\hat{T}_{i-1}+m,\\hat{T}_{i-1}+m+1,\\ldots,\\hat{T}_{i}-m$}\n\t\t\\FOR{$L_i=\\hat{T}_{i-1}+m,\\ldots,\\hat{T}_{i}$}\n\t\t\\STATE Compute $V_i(M_i,L_i)$;\n\t\t\n\t\t\\IF{$V_i(M_i,L_i)\\geq best$}\n\t\t\\STATE  $\\hat{M}_i=M_i, \\hat{L_i}=L_i, V_i(\\hat{M}_i,\\hat{L}_i)=V_i(M_i,L_i)$;\n\t\t\\ENDIF\n\t\t\\ENDFOR\n\t\t\\ENDFOR\n\t\t\\ELSE\n\t\t\\STATE $V_i(\\hat{M}_i,\\hat{L}_i),\\hat{M}_i,\\hat{L}_i$ had been calculated.\n\t\t\\ENDIF\n\t\t\\ENDFOR\n\t\t\\STATE $V_i(\\hat{M}_{\\hat{i}},\\hat{L}_{\\hat{i}})=\\argmax_{0\\leq i\\leq k-1}V_i(\\hat{M}_i,\\hat{L}_i).$\n\t\t\\IF {$V_i(\\hat{M}_{\\hat{i}},\\hat{L}_{\\hat{i}})$ exceeds a threshold,}\n\t\t\\STATE put $\\hat{M}_{\\hat{i}}$ into $\\mathbf{T}$;\n\t\t\\ELSE\n\t\t\\STATE there does not exist new change-point.\n\t\t\\ENDIF\n\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Hierarchical Significance Testing}\n\nHere, we elaborate the use of the bootstrap method mentioned above.\n\nTheorem \\ref{h0} shows that the asymptotic null distribution of  $V(M,T)$ is a mixture of $\\chi^2$ distributions. In practice, it is difficult to directly take advantage of the asymptotic null distribution. So, we use the moving block bootstrap \\citep{kunsch1989jackknife} to obtain the empirical probabilities.\n\nGiven a set of observations $\\{Z_t\\}_{1\\leq t\\leq T}$ and the block size $b_T$, we draw a bootstrap resample $\\{Z_t^*\\}_{1\\leq t\\leq T}$ as follows: (i) define the $b_T$ dimensional vector $X_t=(Z_t,Z_{t-1},\\ldots,Z_{t-b_T+1})$; (ii) resample from block data $\\{X_t\\}_{1\\leq t \\leq T-b_T+1}$  with replacement to get pseudo data $\\{X_t\\}_{1\\leq t\\leq L}$ which satisfies $T=[Lb_T]$, where $[A]$ denotes the integer part of $A$. Denote the first $T$ elements of $\\{X_t\\}_{1\\leq t \\leq L}$ as the bootstrap resample $\\{Z_t^*\\}_{1\\leq t\\leq T}$; (iii) repeat steps (i) and (ii) $R$ times. For the $r$-th repetition, denote the maximum value in equation (\\ref{singleLocLLL}) based on $\\{Z_t^*\\}_{1\\leq t\\leq T}$ by $V(\\hat{M}_1^{(r)},\\hat{L}_1^{(r)})$; (iv) the approximate probability is estimated by $\\{V(\\hat{M}_1^{(r)},\\hat{L}_1^{(r)}):r=1,\\ldots,R\\}.$ Denote the threshold of the estimated probability  by $p_T$, if\n\t $\\frac{\\sharp\\{V(\\hat{M}_1^{(r)},\\hat{L}_1^{(r)})\\geq V(\\hat{M}_1,\\hat{L}_1)\\}}{R+1}<p_T$,  then $\\hat{M}_1$  is a change-point.\n\nIn applications, the choice of the block size $b_T$ involves a trade-off. If the block size becomes too small, the moving block bootstrap will destroy the time dependency of the data and the accuracy will deteriorate. But if the block size becomes too large, there will be few blocks to be used. In other words, increasing the block size reduces the bias and captures more persistent dependence, while decreasing the block size reduces the variance as more subsamples are available. Thus, a reasonable trade off is to  consider the mean squared error as the objective criterion to balance the bias and variance. For the linear time series, as proved in  \\citet{carlstein1986use}, the value of the block size that minimizes MSE is\n$$\nb_T^*=\\left(\\frac{2|\\rho|}{1-\\rho^2}\\right)^{2\/3}T^{1\/3},\n$$\nwhere $\\rho$ is the first order autocorrelation.\nBecause the construction of MSE depends on the knowledge of the underlying data generating sequence, no optimal result is available in  general.\nIn this paper, we follow \\citet{hong2017testing} and \\citet{xiao2007testing} to choose $b_T=\\max\\{q_T,\\bar{q}_T\\}$, where\n\\begin{equation}\\label{bp}\nq_T=\\min\\left\\{\\left[\\left(\\frac{3T}{2}\\right)^{1\/3}\\left(\\frac{2\\hat{\\rho}}{1-\\hat{\\rho}^2}\\right)^{2\/3}\\right],\\left[8\\left(\\frac{T}{100}\\right)^{1\/3}\\right]\\right\\},\n\\end{equation}\nwhere $\\hat{\\rho}$ is the estimator of the first autocorrelation of $Z_t$,\n$\\bar{q}_T$ is the same as (\\ref{bp}) except replacing $\\hat{\\rho}$ with the estimated first order autocorrelation of $Z_t^2$. So the choice of $b_T$ considers the linear dependence and non-linear dependence.\n\n\\subsection{Consistency}\nThe next theorem shows the consistency of the estimated change-point locations under the following assumption.\n\\begin{assumption}\\label{ass1}\nSuppose that $\\mathbf{Z}(T\/0)$  is an absolutely regular sequence which is comprised of two ergodic stationary sequences. Let $\\alpha_1 \\in (0,1)$ denote the fraction of the observations, such that $\\mathbf{Z}(\\lfloor\\alpha_1 T\\rfloor\/0)$ be an ergodic stationary sequence with marginal probability measure $\\mu_1$,  $\\mathbf{Z}(T\/\\lfloor\\alpha_1 T\\rfloor)$ the second ergodic stationary sequence with marginal distribution $\\mu_2$.  Finally, let $\\delta_T$ be a sequence of positive numbers, such that $\\delta_T \\rightarrow 0$ and $T\\delta_T\\rightarrow \\infty$ as $T\\rightarrow \\infty$.\n\\end{assumption}\n\n\\begin{theorem}\\label{asymptotic1}\n\tSuppose Assumption \\ref{ass1} holds. Let $\\hat{M}_1$ be the estimated change-point location from Equation (\\ref{singleLoc}) for a sample of size $T$. For all $\\epsilon>0$ and $T$ large enough such that $\\alpha_1\\in[\\delta_T,1-\\delta_T]$, we have\n\t\\begin{equation*}\n\tP(\\lim_{T\\rightarrow\\infty}|\\frac{\\hat{M}_1}{T}-\\alpha_1|<\\epsilon)=1.\n\t\\end{equation*}\n\\end{theorem}\nThis theorem shows that the consistency only requires the size of each segment  increases to $\\infty$, but not necessarily at the same rate. Under the Assumption \\ref{ass1}, $\\alpha_1$ can be close to 0 or 1 when $T\\rightarrow\\infty$, which is an imbalanced case.\n\nIn the multiple change-points situation, we have the following Assumption.\n\\begin{assumption}\\label{ass2mul}\n\tSuppose that $\\{Z_i\\}_{1 \\leq i \\leq T}$ is an absolutely regular sequence. Let $0=T_0<T_1<\\ldots<T_k<T_{k+1}=T$, and $\\min\\limits_{i=1,\\ldots,k}|T_i-T_{i-1}|\\geq aT^{b}$, with $a>0$ and $0<b\\leq1$. For $i=0,1,\\ldots,k$,  $\\mathbf{Z}(T_i\/T_{i-1})$ is an ergodic stationary sequence with marginal probability measure $\\mu_i$ and $\\mu_i\\neq \\mu_{i+1}$. \\label{key}\n\tFurthermore, let $\\delta_T$ be a sequence of positive numbers, such that $\\delta_T \\rightarrow 0$ and $T\\delta_T\\rightarrow \\infty$ as $T\\rightarrow \\infty$.\n\\end{assumption}\n\n It is worth noting that we do not assume the upper bounds on the number of change-points $k$, but by specifying  the minimum sample size in each segment. In other words, under Assumption \\ref{ass2mul}, as $T\\rightarrow\\infty$, we can have $k\\rightarrow\\infty$ change-points.\n\nAnalysis of multiple change points can be reduced to the analysis of only two change points under Assumption \\ref{ass2mul}.  Let $\\alpha_i=T_i\/T$, for any  $i\\in\\{1,\\ldots,k\\}$. The observations $\\mathbf{Z}(T\\alpha_i\/0)$ can be seen as a random sample from a mixture of probability measures $\\{\\mu_j:j\\leq i\\}$, denoted as $\\boldsymbol{\\mu}_i$. Similarly, observations $\\mathbf{Z}(T\/T\\alpha_{i+1})$ are a sample from a mixture of probability measures $\\{\\mu_j:j\\geq i+1\\}$, denoted here as $\\boldsymbol{\\nu}_i$. The remaining observations are distributed according to some probability measure $\\boldsymbol{\\xi}_i$. Furthermore, $\\boldsymbol{\\mu}_i\\neq \\boldsymbol{\\xi}_i$ and $\\boldsymbol{\\nu}_i\\neq \\boldsymbol{\\xi}_i$.  If one of the previous two inequalities does not hold, we refer to the single change point setting.\n\nConsider any $\\alpha$ such that, $\\alpha_i\\leq\\alpha\\leq\\alpha_{i+1}$. Then, this choice of $\\alpha$ will create two mixture probability measures. One with component probability measures $\\boldsymbol{\\mu}_i$ and $\\boldsymbol{\\xi}_i$, and the other with component probability measures $\\boldsymbol{\\nu}_i$ and $\\boldsymbol{\\xi}_i$. Then, the Ball detection function between these two mixture probability measures is equal to\n\\begin{equation}\\label{multiCP_location}\n\\begin{split}\n&V(\\alpha;\\frac{\\alpha_i}{\\alpha}\\boldsymbol{\\mu}_i+\\frac{\\alpha-\\alpha_i}{\\alpha}\\boldsymbol{\\xi}_i,\\frac{1-\\alpha_{i+1}}{1-\\alpha}\\boldsymbol{\\nu}_i+\\frac{\\alpha_{i+1}-\\alpha}{1-\\alpha}\\boldsymbol{\\xi}_i)\\\\=&\\alpha(1-\\alpha)\\iint_{\\mathrm{A}\\times \\mathrm{A}} [\\frac{\\alpha_i}{\\alpha}\\boldsymbol{\\mu}_i+\\frac{\\alpha-\\alpha_i}{\\alpha}\\boldsymbol{\\xi}_i-\\frac{1-\\alpha_{i+1}}{1-\\alpha}\\boldsymbol{\\nu}_i-\\frac{\\alpha_{i+1}-\\alpha}{1-\\alpha}\\boldsymbol{\\xi}_i]^{2}(\\bar{B}(u,\\rho(u,v))\\omega_{\\alpha} (du)\\omega_{\\alpha} (dv).\n\\end{split}\n\\end{equation}\n\n\\begin{theorem}\\label{lemma_multiCP_max}\nSuppose that Assumption \\ref{ass2mul} holds, then the Ball detection function  in equation (\\ref{multiCP_location}) is maximized when either $\\alpha=\\alpha_i$ or $\\alpha=\\alpha_{i+1}$.\n\\end{theorem}\n\nBy Theorem \\ref{lemma_multiCP_max}, $f_i(\\alpha)$ is maximized when $\\alpha=\\alpha_i$ or $\\alpha=\\alpha_{i+1}$ for $i=1,\\ldots,k-1$. Additionally,\ndefine\n$$V(\\alpha)=\\sum_{i=0}^kV(\\alpha;\\frac{\\alpha_i}{\\alpha}\\boldsymbol{\\mu}_i+\\frac{\\alpha-\\alpha_i}{\\alpha}\\boldsymbol{\\xi}_i,\\frac{1-\\alpha_{i+1}}{1-\\alpha}\\boldsymbol{\\nu}_i+\\frac{\\alpha_{i+1}-\\alpha}{1-\\alpha}\\boldsymbol{\\xi}_i)I(\\alpha_i\\leq\\alpha\\leq\\alpha_{i+1}).$$\nLet $\\mathscr{A}_T=\\{y\\in[\\delta_T,1-\\delta_T]:V(y)\\geq V(\\alpha),\\forall\\alpha\\}$. Let $d(x,\\mathscr{A}_T)=inf\\{|x-y|:y\\in\\mathscr{A}_T\\}$. Then, we have the following Theorem.\n\\begin{theorem}\\label{location_all}\nSuppose that Assumption \\ref{ass2mul}, and $x\\in\\mathbb{R}$,  Let $\\hat{M}_1$ be the estimated change point as defined by equation (\\ref{singleLocLLL}). Then $d(\\hat{M}_1\/T,\\mathscr{A}_T)\\xrightarrow{a.s.}0$ as $T\\rightarrow\\infty$.\n\\end{theorem}\n\nRepeated applications of Theorem \\ref{location_all} can show that as $T\\rightarrow\\infty$, the first $k$ estimated change points will converge to the true change point locations in the manner described above. With a fixed  threshold of the estimated probability $p_T$, all of the change-points will be estimated.\nHowever, with probability approaching 1 as the sample size increases, the number of change-points determined in this way will be more than the true number of change-points, since any given nominal level of significance implies a nonzero probability of rejecting the null hypothesis when it holds. The hierarchical procedure could be made consistent by adopting a threshold for the test that decrease to zero, at a suitable rate, as the sample size increases\\citep{bai1998estimating}.\nThis is illustrated by the following theorem.\n\\begin{theorem}\nLet $\\hat{k}$ be the number of change-points obtained using the hierarchical method based on the statistic (\\ref{kappaformula}) applied with threshold $p_T$, and $k$ be the true number of change-points. If $\\lim\\limits_{T\\rightarrow\\infty}p_T\\rightarrow0$, then under Assumption  \\ref{ass2mul}, $\\lim\\limits_{T\\rightarrow\\infty}P(\\hat{k}=k)=1$.\n\\end{theorem}\nAlthough the hierarchical algorithm tends to estimate more change-points asymptotically when $p_T$ is fixed, this has little effect in practice. For example, the asymptotic probability of selecting $(k+j)$ change-points, is given by $p_T^j(1-p_T),$ which decreases rapidly.\nFurthermore, if there is no change point, that is $k=0$, the probability of selecting at least one change point in our algorithm is\n$$\n\\sum_{j=1}^{\\infty}p_T^j(1-p_T)=p_T.\n$$ Hence the total rate of type I errors is still $p_T$. This is a distinct feature of our hierarchical procedure because controlling for type I errors is a challenging issue in multiple testings.\n\n\n\\section{Simulation studies}\n\nIn this section, we present the numerical performance of the proposed method (BDCP) with $p_T=0.05$ and compare it with several typical methods, including Bayesian method (BCP) \\citep{barry1993bayesian}, WBS method  \\citep{fryzlewicz2014wild}, the graph-based method-gSeg  \\citep{chen2015graph,chu2019asymptotic} and energy distance based method (ECP)  \\citep{matteson2014nonparametric}. BCP, WBS, ECP and BDCP can estimate the number of change-points automatically while gSeg can detect only one change-point or an interval.\n\nThere are four commonly used criteria for the performance of those methods: the adjusted Rand index, the over segmentation error, the under segmentation error and the Hausdorff distance.\nSuppose that the true change-points set is $\\mathbf{T}=\\{0,T_1,\\ldots,T_k,T\\}$ and estimated change-points set is $\\hat{\\mathbf{T}}=\\{0,\\hat{T}_1,\\ldots,\\hat{T}_{\\hat{k}},T\\}$.\nThen denote the true segments of series\n$\\{Z_t\\}_{1\\leq t\\leq T}$ by $\\mathbf{Z}=\\{\\mathbf{Z}(T_1\/0),\\ldots,\\mathbf{Z}(T\/T_k)\\}$ and the estimated segments by\\\\\n$\\mathbf{\\hat{Z}}=\\{\\mathbf{Z}(\\hat{T}_1\/0),\\ldots,\\mathbf{Z}(T\/\\hat{T}_{\\hat{k}})\\}$.\nConsider the pairs of observations that fall into one of the following two sets:\\\\ $\\{S_1\\}=$ \\{pairs of observations in the same segments under $\\mathbf{Z}$ and in same segments under $\\mathbf{\\hat{Z}}$\\};\\\\ $\\{S_2\\}=$ \\{pairs of observations in different segments under $\\mathbf{Z}$ and in different segments under $\\mathbf{\\hat{Z}}$\\}. Denote $\\sharp S_1$ and $\\sharp S_2$ as the number of pairs of observations in each of these two sets. The Rand index $RI$ is defined as\n$$RI=\\frac{\\sharp S_1+\\sharp S_2}{\\binom{T}{2}}.$$\n\nAdjusted Rand index $ARI$ is the corrected-for-chance version of the Rand index which is defined as\n$$ARI = \\frac{RI - E(RI)}{1-E(RI)},$$\nin which 1 corresponds to the maximum Rand index value.\n\nOn the other hand, we also calculate the distance between $\\mathbf{T}$ and $\\hat{\\mathbf{T}}$ by\n$$\\zeta(\\hat{\\mathbf{T}}||\\mathbf{T})=\\sup_{b\\in\\mathbf{T}}\\inf_{a\\in\\hat{\\mathbf{T}}}|a-b|\\ \\ and \\ \\ \\zeta(\\mathbf{T}||\\hat{\\mathbf{T}})=\\sup_{b\\in\\hat{\\mathbf{T}}}\\inf_{a\\in\\mathbf{T}}|a-b|,$$\nwhich quantify the over-segmentation error and the under-segmentation error, respectively \\citep{boysen2009consistencies,zou2014nonparametric}.  The Hausdorff distance  \\citep{harchaoui2010multiple} between $\\mathbf{T}$ and $\\hat{\\mathbf{T}}$ is defined as\n$$\\Delta(\\mathbf{T},\\hat{\\mathbf{T}})=sup\\{\\zeta(\\hat{\\mathbf{T}}||\\mathbf{T}),\\zeta(\\mathbf{T}||\\hat{\\mathbf{T}})\\}.$$\nHere, we only report the results based on adjusted Rand index. The results under other criteria are deferred to the supplementary material.\n\nThree scenarios are used for comparisons: univariate sequence, multivariate sequence and manifold sequence.\nIn each scenario, we consider two types of examples, one without change-point, and one with two change-points as follow:\n$$\\{X_1,X_2,\\ldots,X_n,Y_1,Y_2,\\ldots,Y_m,X_{n+1},X_{n+2},\\ldots,X_{2n}\\}.$$\nThe sample sizes are set to be $n=40$, $m=40,60,80$.\nWe will repeat each model 400 times and the threshold is at 0.05. To save space, some results of univariate sequences are available on the supplementary material.\n\n\\subsection{Multivariate sequence}\nIn this subsection, we consider the $d=3$ dimensional sequences. Examples 4.1.1-4.1.7 are the sequences with no change-point and Examples 4.1.8-4.1.15 are the models with two change-points.\n\\begin{itemize}\n\t\\item \\textbf{Examples 4.1.1-4.1.3:}\n\t$$X_t=\\epsilon_{t},$$\n\t$\\epsilon_{t}\\sim N(0,I_3)$ for Example 4.1.1, $\\epsilon_{t}\\sim t_3(0,I_3)$ for Example 4.1.2 and $\\epsilon_{t}\\sim Cauchy(0,I_3)$ for Example 4.1.3.\n\t\n\t\\item \\textbf{Examples 4.1.4-4.1.5}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$\\epsilon_{t}\\sim N(0,I_3)$ for Example 4.1.4 and $\\epsilon_{t}\\sim t_3(0,I_3)$ for Example 4.1.5.\n\t\n\t\\item \\textbf{Examples 4.1.6-4.1.7}:\n\t$$X_{t}=\\sigma_{X,t|t-1}\\epsilon_{t},$$\n\t$$\\sigma_{X,t|t-1}^{2}=0.02+0.02\\sigma_{X,t-1|t-2}^{2}+0.05X_{t}^{2},$$\n\t$\\epsilon_{t}\\sim N(0,I_3)$ for Example 4.1.6 and $\\epsilon_{t}\\sim t_3(0,I_3)$ for Example 4.1.7.\n\\end{itemize}\n\n\n\\begin{itemize}\n\t\\item \\textbf{Example 4.1.8}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$Y_t=\\mu+0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$\\mu=(4,4,4),(6,6,6),(8,8,8),,\\epsilon_{t}\\sim N(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.9}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$Y_t=\\mu+0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$\\mu=(4,4,4),(6,6,6),(8,8,8),\\epsilon_{t}\\sim t_3(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.10}:\n\t$$X_t=\\epsilon_{t},$$\n\t$$Y_t=\\mu+\\epsilon_{t},$$\n\t$$\\mu=(4,4,4),(6,6,6),(8,8,8),\\epsilon_{t}\\sim Cauchy(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.11}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$Y_t=0.5\\epsilon_{t}\\sigma+0.5\\epsilon_{t-1}\\sigma,$$\n\t$$\\sigma=(3,3,3),(5,5,5),(7,7,7),\\epsilon_{t}\\sim N(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.12}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$Y_t=0.5\\epsilon_{t}\\sigma+0.5\\epsilon_{t-1}\\sigma,$$\n\t$$\\sigma=(3,3,3),(5,5,5),(7,7,7),\\epsilon_{t}\\sim t_3(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.13}:\n\t$$X_t=\\epsilon_{t},$$\n\t$$Y_t=\\epsilon_{t}\\sigma,$$\n\t$$\\sigma=(9,9,9),(16,16,16),(25,25,25),\\epsilon_{t}\n\t\\sim Cauchy(0,I_3).$$\n\t\n\t\\item \\textbf{Examples 4.1.14-4.1.15}: \\\\\n\t$X_t,Y_t \\sim CCC-GARCH(1,1)$\\citep{Bollerslev1990Modelling}, let\n\t$$X_t=\\sigma^{X}_{t}\\epsilon_t,Y_{t}=\\sigma^{Y}_{t}\\epsilon_{t},$$\n\t$$\\sigma^{X}_{t}=(\\omega_{X}+A^{X}\\epsilon_{t-1}+B_{X}\\sigma{X}_{t-1}^{2})^{1\/2},$$\n\t$$\\sigma^{X}_{t}=(\\omega_{Y}+A^{Y}\\epsilon_{t-1}+B_{Y}\\sigma{Y}_{t-1}^{2})^{1\/2}.$$\nLet     $\\omega^{X}=(0.01,0.01,0.01),$ $A^{X}=diag(0.02,0.03,0.01),$ $B^{X}=diag(0.02,0.02,0.05)$, and\n\n\tCase 1:$$\\omega^{Y}=2\\omega^{X},A_{Y}=4A^{X},B^{Y}=5B^{X},$$\n\t\n\tCase 2:$$\\omega^{Y}=3\\omega^{X},A_{Y}=5A^{X} ,B^{Y}=6B^{X}.$$\n\t\n\tCase 3:$$\\omega^{Y}=4\\omega^{X},A_{Y}=6A^{X} ,B_{Y}=7B^{X}.$$\n\t$\\epsilon_{t}\\sim N(0,1)$ for Example 4.1.14 and $\\epsilon_{t}\\sim t(df=3)$ for Example 4.1.15.\n\\end{itemize}\n\n\nTable \\ref{Sim_adj_mul_type1} reveals that ECP and BDCP can handle the multivariate stationary series well. BCP works well when the distribution is normal but has a lower adjusted Rand index in $t$ distribution and Cauchy distribution. We do not consider the WBS method because it can not handle the multivariate cases. Tables \\ref{Sim_adj_mul_mean}-\\ref{Sim_adj_mul_gar} present the results of those examples with two change-points. All four methods have excellent performance in the multivariate normal distribution and multivariate $t$ distribution with location shift.\nExamples 4.1.11-4.1.13 consider the scale shift case and Examples 4.1.14-4.1.15 are the popular GARCH models which are also the scale shift case. We can see from Tables  \\ref{Sim_adj_mul_var}-\\ref{Sim_adj_mul_gar} that BDCP has the best performance in almost all the scale shift cases.\n\n\n\\subsection{Manifold-valued sequence}\nIn this subsection, we report some manifold-valued examples where ECP can not detect the change-points but BDCP works well. Consider the distribution in a unit circle and let\n$$P_1 \\sim Unif([-\\pi\/6,\\pi\/6)\\bigcup[11\\pi\/6,13\\pi\/6)),$$\n$$P_2 \\sim Unif([\\pi\/3,2\\pi\/3)\\bigcup[7\\pi\/3,8\\pi\/3)),$$\n$$P_3 \\sim Unif([5\\pi\/6,7\\pi\/6)\\bigcup[17\\pi\/6,19\\pi\/6)),$$\n$$P_4 \\sim Unif([4\\pi\/3,5\\pi\/3)\\bigcup[10\\pi\/3,11\\pi\/3)),$$\n$$P_5 \\sim Unif([0,4\\pi)).$$\nWe calculate the circular distance which is defined by\n\\begin{equation}\\label{circlediatance}\nd(X_i,X_j)=min(|X_i-X_j|,2\\pi-|X_i-X_j|).\n\\end{equation}\nWe simulate four examples with 0,1,2,3 change-points respectively. Let $n=40, m=40,60,80$.\n\\begin{itemize}\n\t\\item \\textbf{Example 4.2.1}:\n\t$$\\{X_1\\ldots,X_{3n}\\}\\sim P_5.$$\n\t\\item \\textbf{Example 4.2.2}:\n\t$$\\{X_1\\ldots,X_n\\}\\sim P_1, \\{Y_1\\ldots,Y_m\\}\\sim P_3.$$\n\t\n\t\\item \\textbf{Example 4.2.3}:\n\t$$\\{X_1\\ldots,X_n\\}\\sim P_1, \\{Y_1\\ldots,Y_m\\}\\sim P_3,\\{X'_1\\ldots,X'_n\\}\\sim P_2.$$\n\t\n\t\\item \\textbf{Example 4.2.4}:\n\t$$\\{X_1\\ldots,X_n\\}\\sim P_1, \\{Y_1\\ldots,Y_m\\}\\sim P_3,$$\n\t$$\\{X'_1\\ldots,X'_n\\}\\sim P_2, \\{Y'_1\\ldots,Y'_m\\}\\sim P_4.$$\n\\end{itemize}\n\n\n\\begin{comment}\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=1\\textwidth]{sim_exp.pdf}\n\\caption{\\label{sim_exp}}\n\\end{figure}\n\\end{comment}\n\n\nTable \\ref{Sim_adj_man_type1} reveals that BCP, WBS, ECP all perform well when there is no change-point. However, they do not work when the sequences have change-points (Table \\ref{Sim_adj_man_123}). That is because BCP is based on the normal distribution, and WBS is a CUSUM statistics which does not work in a circular distribution. For ECP, that is because the circular distance is not of strong negative type (Theorem 9.1 in \\citet{hjorth1998finite}).  The gSeg method can detect change-points when the number of change-points is one or two but do not perform well in Example 4.2.4. BDCP has a remarkable performance in all these examples.\n\n\\section{Real data analysis}\n\n\n\\subsection{Wind direction of Yunnan-Guizhou Plateau}\n\nMonsoon is used to describe seasonal changes in atmospheric circulation and precipitation associated with the asymmetric heating of the land and sea. The major monsoon systems in the world consist of West African Monsoon (WAM), Indian summer monsoon (ISM), East Asian Monsoon (EAM) and so on. In this subsection, we analyze the wind direction data of Yunnan-Guizhou Plateau ($105^\\circ$E, $27^\\circ$N) from 06\/01\/2015 to 10\/30\/2015. The data are available in R package \\emph{rWind}. Yunnan-Guizhou Plateau is located in southwest China, with local climate influenced by both ISM and EAM  \\citep{sirocko1996teleconnections}\\citep{li2014synchronous}(Fig. 1A and S1). Strict spatial boundaries between the ISM and the ASM are difficult to define \\citep{cheng2012global} though previous researchers have suggested $103^\\circ$E as the dividing line on the basis of summer prevailing winds.\n\nDaily wind directions are shown in the top-left of Figure \\ref{wind1}. Note that degree 0 represents due North, $\\pi\/2$ represents due East, $\\pi$ represents due South and $3\\pi\/2$ represents due West.\nWe can see that the wind directions are distributed in almost all directions. In the beginning, the most widely distributed direction is the southwest wind from the Indian Ocean, and then turns smoothly to southeast, which is from the Pacific Ocean. In particular, Yunnan-Guizhou Plateau was mostly influenced by ISM in June and July. After July, the influence of EAM gradually increased  \\citep{li2015introduction}.\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{guizhou_2015_1_20_27_105.pdf}\n\t\\caption{\\label{wind1}Wind direction of Yunnan-Guizhou Plateau from 06\/01\/2015 to 10\/30\/2015 and the performance of BCP, WBS, gSeg, ECP and BDCP. The y-axis shows wind direction. Degree 0 and $2\\pi$ represent due North, $\\pi\/2$ represents due East, $2\\pi$ represents due South and $3\\pi\/2$ represents due West.}\n\\end{figure}\n\nTo detect the change-point in the wind direction series, we calculate the circular distance between the daily direction as defined in (\\ref{circlediatance}).\n\nThe performance of the five methods is shown  in Figure \\ref{wind1}.\nBCP, WBS and ECP can not detect any change-point, as seen in the simulation studies in subsection 4.2. gSeg detects an interval between ``07\/17\/2015'' and ``07\/28\/2015''. BDCP estimates four change-points located at ``07\/01\/2015'', ``07\/29\/2015'', ``09\/02\/2015'' and ``09\/26\/2015''.\n\n\nTo visualize the result of BDCP, Figure \\ref{five_periods} depicts the wind rose plot for the five periods detected by BDCP. We can see the significant changes of the direction distribution especially between 07\/29\/2015 - 09\/01\/2015 and 09\/02\/2015 - 09\/25\/2015. The wind directions are almost southwest or west in June and July, then  turn to southeast in September \\citep{li2015introduction}. As mentioned in \\citet{hillman20178}, 75\\% of the average annual precipitation falls in the months of June-September associated with the ISM, and the ISM gets weaker during June and July because isolation decreases by 2-3\\%. BDCP can perfectly detect the change of influence between ISM and EAM.\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{five_periods.pdf}\n\t\\caption{\\label{five_periods}The wind rose plot for the five periods detected by BDCP.}\n\\end{figure}\n\n\n\\subsection{Bitcoin price}\nBitcoin is the most popular form of cryptocurrency in recent years.  According to research of Cambridge University in 2017  \\citep{hileman2017global}, there are 2.9 to 5.8 million unique cryptocurrency wallet users, most of whom use Bitcoin. One of the known features of Bitcoin is its high volatility.\nBitcoin is not a denominated flat currency and there is no central bank overseeing the issuing of Bitcoin,  its price is thus driven solely by the investors. Using the weekly data over 2010-2013 period,   \\citet{briere2015virtual} showed that Bitcoin investment had some high distinctive features, including exceptionally high average return and volatility. Hence, accurately fitting its variation is important \\citep{chu2015statistical}.\n\n\nBitcoin can be exchanged for other currencies, products, and services in legal or black markets. \\citet{chu2015statistical} measured the volatility of Bitcoin exchange rate against six major currencies. They found that the behavior of Bitcoin was sharply different from those currencies; its interquartile range was much wider, its skewness was much more negative, its kurtosis was much more peaked and its variance was much larger. Bitcoin showed the highest annualized volatility of percentage change in daily exchange rates. In this subsection, we detect the change-points of daily log-return of Bitcoin using methods, BCP, WBS, gSeg, ECP and BDCP. The datasets are available on \\url{http:\/\/api.bitcoincharts.com\/v1\/csv\/bitstampUSD.csv.gz}. Figure \\ref{bitcoin} displays the  exchange rate of Bitcoin and daily log-returns during 09\/13\/2011 - 12\/31\/2012.\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{bitcoin_price.pdf}\n\t\\caption{\\label{bitcoin} US dollar-Bitcoin exchange rate and daily log-returns from 09\/13\/2011-12\/31\/2012.}\n\\end{figure}\n\nFigure \\ref{bitcoincp} compares the performance of the five methods. BCP and WBS can not handle the severe volatility at the beginning of the sequence.\ngSeg detects one change-point at ``02\/23\/2012'', and ECP estimates a change-point at ``02\/09\/2012''.\nBDCP detects four change-points at ``02\/11\/2012'', ``04\/16\/2012'', ``05\/19\/2012'', and ``08\/21\/2012''.\n\nOn February 11, 2012, Paxum, an online payment service and popular means for exchanging Bitcoin announced it would cease all dealings related to the currency due to the concerns of its legality. Two days later, regulatory issues surrounding money transmission compelled the popular Bitcoin exchange and service firm TradeHill to terminate its business and immediately began selling its Bitcoin assets to refund its customers and creditors. Bitcoin trading started to cool down during that period.\n\nAfter May 19, the price of Bitcoin had increased from \\$5.07 to the maximum \\$14.14 on August 17 and kept at that level after that. The reasons for the rise were many. Lots of online articles on this subject expressed the same message: Bitcoin was now going mainstream. WordPress, ranked by Alexa as the 21st most popular site in the world, started to accept Bitcoin for payment on November, 2012.\n\nThe variances of these five stages detected by BDCP are: 0.1054, 0.0182, 0.0059, 0.0458, and 0.0177. The daily log-return sequence was very flat and the price almost did not change during period 04\/16\/2012 - 05\/18\/2012. But it was volatile during other periods from Figure \\ref{bitcoincp}.\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth,height=0.9\\textheight]{bitcoin_cp2.pdf}\n\t\\caption{\\label{bitcoincp} The results of BCP, WBS, geg, ECP, BDCP on the daily log-returns of Bitcoin series from 09\/13\/2011-12\/31\/2012.  The y-axis shows daily log-returns of Bitcoin series. The red lines are the change-point locations detected by BCP, WBS, geg, ECP, BDCP. } \n\\end{figure}\n\n\n\\section{Conclusion}\nWe developed a change-point detection procedure for weakly dependent sequences. Our key idea lies in the novel measure of Ball detection function.\nWe proved the asymptotic properties of its sample statistic for absolutely regular sequences.  Extensive simulation studies demonstrated that our method had a superior performance to other existing methods in various settings. Two real data analyses indicated that our method was useful in analyzing non-Euclidean sequences with various change points and led to insightful understanding of the data. Also, our method is robust since our test statistic is rank-based.\n\nWe will further investigate Ball detection function and its related concepts. For example,  the current computational complexity of our proposed algorithm is $O(kT^2\\log T)$, where $k$ is the number of change-points, and $T$ is the length of the sequence. It will be useful to find an algorithm with a lower computational complexity.\n\n\\bibliographystyle{ECA_jasa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn $1935$ Einstein and Rosen \\cite{ER} proposed the metric\n\\begin{eqnarray}\\label{ER}\n&ds^2 =  -\\frac{u^2}{u^2+2m}dt^2 + 4(u^2+2m) du^2+(u^2+2m)^2( d\\theta^2 + \\sin^2 \\theta \\, d\\varphi^2) \\\\\n&-\\infty < u < 0 \\, ,  0 < u < \\infty \\ ,\n\\end{eqnarray}\nobtained in two steps, first changing the Schwarzschild radial coordinate $ \\rho = u^2 + 2m, \\, u > 0$ and then, allowing the new  coordinate to take also values  in $-\\infty < u < 0 $, producing two copies of the exterior Schwarzschild metric. But this  metric is degenerate, $ det ( g_{\\mu\\nu})=0$ at $u=0$, and the absence of good coordinates to cover the union of the two exterior spaces was not considered.\\\\\n \nRecently, Katanaev \\cite{KATA1}  obtained this metric as a solution of Einstein's equations with a $\\delta$-type energy momentum tensor corresponding to a point particle at $ r= 0$ in isotropic coordinates. \n But he  pointed out  the incompleteness of its solution, in the same sense as in the original Einstein-Rosen metric, because he did not solve the problem of the coordinate singularity at $u=0$.  Despite that, he demonstrated the complete bridge passability in \\cite{KATA2}. \nGuendelman et al.  \\cite{GUEN1} claimed that the correct interpretation of the incomplete original Einstein-Rosen bridge has as source  a generalized function (a distribution)  with support on the tridimensional bridge, interpreted as a light-like thin shell (L-L brane). Using Eddington-Finkelstein-like coordinates they obtained an \n extension with discontinuous first derivatives in the metric, producing the distributional character of the source. The complete passability  of  the bridge, in both senses, and the existence of closed time-like geodesics was discussed in \\cite{GUEN2}.\\\\\n \nWe must contemplate the notion of extension of  a space-time, and consider \nthe possibility of the existence of many different extensions. One can get a maximal extension of a metric, in the sense that all geodesics are complete, but there can exist more than one maximal extension (see for example the nice book of Earman \\cite{EARM}).  In fact,   Katanaev commented in \\cite{KATA1} that he had to choose  between two possible solutions  for the  lapse function, and that having taken the discarded one he would have obtained the same conclusion as Guendelman et al. \\\\\n\n\nThe purpose of this work is to study the possibility of other extensions of the Einstein-Rosen space-time. To find them, it will be useful to take into account  that any two-dimensional metric (in our case the metric of the submanifolds $ \\theta , \\varphi $ constant), verifying some specific conditions, admits isothermal coordinates. So in section \\ref{ERBSec} we propose a method for constructing isothermal coordinates, that will be systematically used  to find all possible extensions of the original Einstein-Rosen bridge. We shall obtain first  the  Einstein-Rosen bridge with boundary ($ERb$), which is a non maximal extension of the original Einstein-Rosen metric.  It is also a subspace of the Kruskal-Szekeres space-time \\cite{KRUS}, which  has  been  considered as the  maximal extension of the original Einstein-Rosen bridge, but it is not the  only possibility. In section \\ref{maximal} we show two other possible extensions by doing a change of topology. One of them, according to Poplawski \\cite{POPLA}, should coincide with the aforementioned Guendelman et al.  extension, so we will study the other one. We call it the hyperbolic Einstein-Rosen bridge ($hER$), since it is diffeomorphic to the covering space of an hyperboloid, as we prove in Appendix A. In section \\ref{ED} we provide this extension with a differential structure, and obtain its corresponding metric in section \\ref{mher}. We find that in this extension the bridge is generated by isotropic geodesics and it is traverable, and furthermore, the first derivatives of the metric are continue on it, in contrast to the Guendelman's extension. However, the metric presents an unexpected singularity in one side, but, as in the Schwarzschild metric, it is concealed by a horizon, which in this case is the bridge itself. Finally, in section \\ref{source} we discuss which kind of physical process could generate our extension (or a part of it), and in section \\ref{conclusions} we summarize the main conclusions. In order to facilitate the reading, in Appendix B we gather the main definitions used in the paper.\n\n\\section{The Einstein-Rosen bridge with boundary $ERb$}\\label{ERBSec} \n\n{\\bf{Proposition 1}}: Let us consider a two-dimensional metric  of the  type $ ds^2 = g_{_{11}}(\\varphi_2) d \\varphi_{_{1}}^2 + g_{_{22}}(\\varphi_2) d \\varphi_{_2}^2$, with $ g_{11}g_{22} < 0$.\n Under the change of coordinates\n\\begin{eqnarray}  \n&& U = f(\\varphi_2) \\cosh \\varphi_{_1} \\, ,   V = f(\\varphi_{_2}) \\sinh \\varphi_{_1}  \\, ,\\mathrm{ if }\\,  \\    g_{22} > 0\\\\\n&& U = f(\\varphi_2) \\sinh \\varphi_{_1} \\, ,   V = f(\\varphi_{_2}) \\cosh \\varphi_{_1} \\, ,\\mathrm{ if } \\,  \\    g_{22} < 0 \\ ,\n\\end{eqnarray}\nit  may be written as $ ds^2 = \\Omega^2 ( - dV^2 + d U^2)  $ with : $ \\Omega^2 = \\frac{\\mid g_{_{11}}\\mid}{f^2} \\, ,\\  \\     f = C e ^{\\pm \\int \\sqrt{\\frac{ \\mid g_{_{22}}\\mid}{\\mid g_{_{11}}\\mid} }d\\varphi_{_2}}$, and $C$ a non null arbitrary constant.\\\\\n\nThe proof is simple. A straightforward calculation transforms the second form of the metric into the first form. \nLet us  apply this result to the metric  $ ds^2 =  -\\frac{u^2}{u^2+2m}dt^2 + 4(u^2+2m) du^2$ of the \n$t$-$u$  section of the  Einstein-Rosen bridge.\nWe must identify in this case  $\\varphi_1 = t\/k$, taking for $k$ a positive constant to be determined bellow, and  $ \\varphi_2 = u$, and substitute $g_{11}= -\\frac{k^2 u^2 }{u^2+2m}$, $ g_{22}= 4(u^2+2m)$.\nThe change of coordinates $ U = f_{_{E R}}(u) \\cosh (\\frac{t}{\\kappa}) \\, , \\    V = f_{_{E R}}(u) \\sinh (\\frac{t}{\\kappa})$ produces\n\\begin{eqnarray}\n& f_{_{E R}}= C e^{\\pm\\frac{u^2}{\\kappa}}u^{\\pm\\frac{4m}{\\kappa}}\\\\\n& \\Omega^2= \\frac{\\kappa^2}{C^2}\\frac{u^2}{u^2+2m} \\frac{1}{e^{\\pm\\frac{2u^2}{\\kappa}}u^{\\pm\\frac{8m}{\\kappa}}}\n\\end{eqnarray}\nin the region $ u > 0$, and we can extend it to the region $u<0$. Taking  $\\kappa = 4m$ and choosing sign $(+)$ we avoid the metric degeneration, and we get\nin the region  $ U^2-V^2 \\geq 0 $ the metric  \n\\begin{eqnarray}\n& ds^2 =\\frac{16 m^2}{C^2}\\frac{e^{-\\frac{u^2}{2m}}}{u^2+2m}\\left(- d V^2 + dU^2\\right)\\label{ERI}\\\\\n& U^2-V^2 =  C^2 e^{\\frac{u^2}{2m}}u^2\\label{E} \\ .\n\\end{eqnarray}\nThe second equation defines a monotonous  function  $u^2 = h_{ER}(U^2-V^2)$ of the variable $U^2-V^2$, that is derivable in the set $ERb= \\{(U,V)\\in \\mathbb{R}^2 \\mid U^2-V^2 \\geq 0\\} - (0,0)$. \nWe will use below  the inverse change given by $ u=\\mathrm{sig} (U)\\sqrt{h_{ER}(U^2-V^2)}  $.   The set $ U^2 - V^2 = 0 $ is the topological  boundary of the open  $ U^2 - V^2 > 0$, and corresponds to the coordinate axis  $u=0\\, (r=2m)$.\\\\\n\nThe constant $C$ may be determined in order to reach the exterior part of the Schwarzschild metric, obtaining $C= 1\/\\sqrt{2m}$. Using the $h_{ER}$ function defined above we can express the metric in the form\n\\begin{eqnarray}\n&  ds^2 =\\Omega^2_{ER} \\left(- d V^2 + dU^2\\right)\\label{ERI1}\\\\\n& \\Omega^2_{ER}= \\frac{32m^3}{h_{ER}(U^2-V^2)+2m}e^{-\\frac{h_{ER}(U^2-V^2)}{2m}}\n\\label{ERI2}  \\ .\n\\end{eqnarray}\n As in the Kruskal-Szekeres' extension of the Schwarzschild metric (podriem posar la refer\u00e8ncia de Kruskal \\cite{KRUS}), the isothermal coordinates  have solved the trouble at $r=2 m$, but in this case the extension does not cover the region $ U^2-V^2 < 0$. The set of points $ERb= \\{(U,V)\\in \\mathbb{R}^2 \\mid U^2-V^2 \\geq 0\\} - (0,0)$,  is a manifold with boundary  $ \\partial ERb = \\{ (U,V)\\in \\mathbb{R}^2  \\mid U^2-V^2 =0\\}-(0,0)$. \n The metric is well defined and derivable on $ERb$, and the boundary just corresponds to  the bridge, but it is a bridge to nowhere.\nThis manifold   is properly an extension of the Einstein-Rosen space, because it adds points to the space that were not covered by the Einstein-Rosen  coordinates. We shall call this extension the Einstein-Rosen bridge with boundary ($ERb$). If we stop here we would have a serious drawback because the radial isotropic geodesics   $ U+V= \\mathrm{ const. } \\, , \\  \\  U - V = \\mathrm{ const. } $  would be incomplete, as we show in Fig.\\ref{ERB}.\n\n\\section{Possible maximal extensions}\\label{maximal}\n\\subsection{The Kruskal-Szekeres space-time}\\label{KS}\nIt is trivial to consider the previous extension as a subspace of the Kruskal-Szekeres space-time. In this way  the geodesic incompleteness  at the boundary disappears, but one gets   the singularity at $r=0$. Moreover, only space-like curves might connect the two isometric infinite spaces of the bridge, through the two-dimensional region $\\{ U=V=0$ and any angles $\\theta, \\varphi \\}$. Therefore, this extension of the Einstein-Rosen bridge is not traversable.\n  Usually, the original Einstein-Rosen metric is extended in this sense \\cite{MTW}, but this is not the only possiblity, so we come back to our manifold with boundary to look for alternative extensions. \n\\subsection{Two more extensions by changing the topology }\\label{free}\nIt is well manifest that the metric given in  (\\ref{ERI1}) and (\\ref{ERI2}) is the same over all the points of the boundary, then we can change the topology by identifying points of it, without come into terms with the field equations. (Let us recall a precedent in the sixties, when W. Rindler \\cite{RIND} identified the points $ \\{U,V, \\theta, \\varphi\\}$ and $\\{ -U,-V, \\theta, \\varphi\\}$ of the Kruskal-Szekeres space-time to construct the Elliptic Kruskal-Schwarzschild space-time. It has the virtue of giving, in the limit $m = 0$, the Minkowski space-time instead of two copies of it, as it happens in the K-S space).\n In principle, as we show in Fig. \\ref{ERB}, we have two possibilities:\nwe can identify points with the same coordinate $V$, i.e; pairs of points as $A$ and  $B$, and pairs as $C$ and $D$, or else we  can identify  points symmetric with respect to the center $(0,0)$, i.e; identify pairs as $A$ and $D$ and pairs as $B$ and $C$. The  change of topology may help to extend the space-time.\\\\\n\n \\begin{figure}[hbt]\\label{ERBcopy}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{ERbOriented.pdf}\n\\caption{\\small{Diagram of the $ERb$ extension (where the origin $(0,0)$ is excluded), showing how the incompleteness of the  isotropic geodesics (red lines) may be in principle avoided with a change of topology by gluing points $ A\\equiv B , C\\equiv D $ , and choosing the appropriated light cones orientations.}}\\label{ERB}\n\\end{figure}\n\nIn the next section {\\bf{we shall use the first identification of points of the $ERb$ manifold ($ A\\equiv B\\, , \\, C\\equiv D$)}},  and    provide the quotient space $ERb\/\\hspace{-0.17 cm}\\sim$   with a differentiable structure, obtaining what we have called the hyperbolic Einstein-Rosen bridge.\nThe set formed by all the identified points  defines {\\bf{ the bridge $B$ connecting}}  the two sheets  of the Einstein-Rosen space. The other possible identification of points  has been commented by Poplawski \\cite{POPLA}\nand associated to the extension of the Einstein-Rosen  bridge described by Guendelman et al. \\cite{GUEN1}, which has  a $LL$-light brane installed in the  bridge. \\\\\n\nThe following two remarks will help to understand the rest of the paper. Firstly, we  must  realize that   the manifold $ERb\/\\hspace{-0.19 cm}\\sim$  obtained by gluing points will have a different differentiable structure than $ERb$. This means that the   coordinates to be introduced  in a neighbourhood of the bridge $B$ cannot be linked to the coordinates of the $ERb$ manifold of section \\ref{ERBSec} by a diffeomorphism. The second remark refers to the orientation of the light cones. Since we are going to identify events $A$ and $B$, if we choose in the region $U > 0$ the orientation down-up for the light cone (as shown in Fig. \\ref{ERB}), then  {\\bf{ we must choose the  up-down orientation  in the region $U < 0$}}, in order  to make topologically possible that a  light ray can pass through the bridge.\n\n\n\\section{A differentiable structure for the quotient space $ERb\/\\hspace{-0.17 cm}\\sim$}\\label{ED}\n\nIn order to give a differentiable structure to the quotient space  $ERb\/\\hspace{-0.17 cm}\\sim$ considered in \\ref{free},\n   we must introduce a system of  compatible coordinates covering all  the space.  We have got it  with only a pair of coordinate systems  that convert  $ERb\/\\hspace{-0.17 cm}\\sim$  into a manifold  diffeomorphic to the covering space of an hyperboloid of revolution  with signature $(- \\ +)$. We deviate to the Appendix A the  description of this covering space in  isothermal coordinates, denoted as $(\\bar U , \\bar V)$, whose values define the set $\\bar H = \\{(\\bar U , \\bar V ) \\mid  \\bar U^2 - \\bar V^2 > 0 \\, , \\bar U > 0\\} $ showed in the right hand of Fig. \\ref{cordenadesU1V1}-\\ref{entorncopy1}. The throat of the hyperboloid generates in the covering space the hyperbola\n  $\\bar J =\\{(\\bar U,\\bar V)\\mid  \\bar U^2 - \\bar V^2= D^2\\, , \\bar U > 0\\}$, which will correspond to the bridge.  \n  We start by introducing a map \n $P_1: \\Gamma_1 \\subset ERb \\, \\rightarrow  \\bar H_1\\subset \\bar H $, defined on the interior of $ERb$:  $\\Gamma_1 = \\{(U,V)\\in \\mathbb{R}^2 \\mid U^2-V^2 > 0\\}$,\nand giving values over the covering of the hyperboloid minus the covering of its throat: $ \\bar H_1=\\{(\\bar U_1,\\bar V_1)\\mid  \\bar U_1^2 - \\bar V_1^2  >  0 \\, , \\bar U_1 > 0\\} \\setminus\\bar J$. We define it as follows\n\\begin{eqnarray}  \\label{co1}\n &P_1(U,V)=(\\bar U_1 , \\bar V_1) \\\\\n &\\bar U_1 =  DS\\left(\\mathrm{sig} (U) \\sqrt{h_{ER}(U^2-V^2)}\\right)\\frac{\\mid U \\mid}{\\sqrt{U^2-V^2}}\\\\\n &\\bar V_1 =  DS\\left(\\mathrm{sig} (U) \\sqrt{h_{ER}(U^2-V^2)}\\right)\\frac{V}{\\sqrt{U^2-V^2}} \\ ,\n\\end{eqnarray}\nwhere $D$ is an arbitrary positive constant and $S(\\cdot)$ is a  positive, smooth and monotonous increasing function verifying $S(0)=1$. The values of this function allow the {\\bf{introduction of the coordinate system   $(\\bar U_1 , \\bar V_1)$}} over the subset $\\Gamma_1\\subset ERb\/\\hspace{-0.17 cm}\\sim$. In Appendix A we prove that, with this definition, the assignation $(\\bar U_1,\\bar V_1)=(\\bar U,\\bar V)$ is a diffeomorphism between our manifold minus the bridge and the covering of the hyperboloid minus the throat\n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{U1V1.pdf}\n\\caption{\\small{Map $P_1: \\Gamma_1\\rightarrow \\bar H_1$. It introduces the coordinates $(\\bar U_1 , \\bar V_1)$ over $ERb\/\\hspace{- 0.17 cm}\\sim$  excluding the bridge, which is represented as an hyperbola (with point $(D,0)$ removed) in these coordinates.}}\\label{cordenadesU1V1}\n\\end{figure}\n\nTo complete the differentiable  structure we need coordinates for a neighbourhood of the bridge $B$.\nTo this end, we consider the map $P_2: \\Gamma_2 \\subset ERb \\rightarrow  \\bar H_2 \\subset \\bar H$, with  $\\Gamma_{2}= \\{(U,V)\\mid  0 \\leq    U^2 -V^2 < \\epsilon^2 \\} - (0,0)$, defined as\n \\begin{equation} \\label{U2V22}\n P_2(U,V)=(\\bar U_2 , \\bar V_2) = \\left( \\sqrt{V^2+D^2}+\\frac{U^2-V^2}{U}\\, \\ , V\\right) \\ \n\\end{equation}\nThe bridge $B$ is mapped onto $\\bar J -(D,0)$,  where $\\bar J $ is the hyperbola defined above. The image set $\\bar H_2= P_2(\\Gamma_2)$ is now a neighborhood of the set $\\bar J -(D,0)$, defined as\n \\begin{eqnarray}\n  & \\bar H_2 =   \\{(\\bar U_2,\\bar V_2)\\mid   \\bar U_{L}(\\bar V_2) < \\bar U_2 < \\bar U_{R}(\\bar V_2)  \\} -(D,0)\\label{neighborhood} \\\\\n & \\bar U_{L}(\\bar V_2)= \\sqrt{\\bar V_2^2 + D^2 }-  \\frac{\\epsilon^2}{\\sqrt{\\bar V_2^2+\\epsilon^2}}\\, \\ ,  \\ \n  \\bar U_{R}(\\bar V_2)= \\sqrt{\\bar V_2^2 + D^2 }+ \\frac{\\epsilon^2}{\\sqrt{\\bar V_2^2+\\epsilon^2}} \\ ,\n \\end{eqnarray}\n  as shown in the right hand of Fig.  \\ref{entorncopy1}. This map  verifies $ P_2(-b,b)= P_2(b,b)$ and $ P_2(-b,-b)= P_2(b,-b)$ for any $b >0$, i.e, it is not injective over the boundary $ \\partial ERb$ defined in \\ref{ERBSec}.  The set $\\Gamma_2 \\setminus \\partial ERb $ is formed by two disjoint open subsets $ \\{R_i\\}_{ i= 1,2}  \\subset \\Gamma_2 \\setminus \\partial ERb$, and $P_2$ maps them bijectively onto two open subsets $ \\{\\bar R_i\\}_{ i= 1,2}  \\subset  \\bar H_2\\setminus\\ \\bar J $, i.e,   $P_{2}( R_i) = \\bar R_i$ as shown in Fig. \\ref{entorncopy1}. \n We shall need the coordinates of the sets $R_i = P_{2}^{-1}( \\bar R_i)$ as functions of the coordinates of $\\bar R_i$.   One gets  \n \\begin{eqnarray}\n && P_{2}^{-1}(\\bar R_1)= \\{( U_{-}, V)\\}\\, \\ \\ , \\, \\ \\  P_{2}^{-1}(\\bar R_2)= \\{(U_{+},V)\\}\\label{pedos}\\\\\n && U_{+} = \\frac{1}{2}(\\sqrt{\\varphi^2 +4 \\bar V_2^2} - \\varphi ) \\,  \\   , V=\\bar V_2 \\label{antecedent+}\\\\\n&& U_{-} = -\\frac{1}{2}(\\sqrt{\\varphi^2 +4 \\bar V_2^2} + \\varphi ) \\,  \\   , V=\\bar V_2 \\, \\ , \\ \\varphi = \\sqrt{\\bar V_2^2+ D^2}-\\bar U_2\\label{antecedent-} \\ .\n \\end{eqnarray}\nThe non-injectivity of $P_2$ over the boundary $ \\partial ERb$\n will make possible  to assign the same coordinates to pairs of points like  $A , B $ and  $C , D$. By identifying the pairs  $(-b,b) , (b,b)$ and  $(-b,-b) , (b,-b)$ of the set $\\Gamma_2, \n$ shown in the left part of Fig. \\ref{entorncopy1}, we obtain a neighborhood of the bridge of the quotient space $\\mathrm{ERb} \/\\hspace{-0.17 cm}\\sim$.\\\\\n\n \\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{P2_3.pdf}\n\\caption{ Map $P_2 :  \\Gamma_{2}\\rightarrow \\bar H_2$. It maps   a neighborhood of the bridge $B$, formed by the set of identified points,  onto a neighborhood of the  covering of the throat of the hyperboloid, represented by the hyperbola in green.  The values of this function define the coordinates  $( \\bar U_2 , \\bar V_2)$.}\\label{entorncopy1}\n\\end{figure}\n\nWith this application we can proceed to the {\\bf{introduction of the coordinate system $(\\bar U_2 , \\bar V_2)$}}\nin a  neighborhood $\\Gamma_2\/\\hspace{-0.17 cm}\\sim \\, \\ \\subset ERb \/\\hspace{-0.17 cm}\\sim$  of the bridge of  the quotient space  as follows. To the pair of  points $ (-V,V) \\equiv (V,V)$ we assign  coordinates $(\\bar U_2 , \\bar V_2) = ( \\sqrt{ V^2+ D^2}, V)$. To  points such that  $U^2 - V^2 > 0$ we assign coordinates  $(\\bar U_2 , \\bar V_2) = (\\sqrt{V^2 + D^2} +\\frac{U^2-V^2} {U}, V)$. With the next proposition we shall provide a differentiable structure to the quotient space  $ERb \/\\hspace{-0.17 cm}\\sim$. \\\\\n  \n{\\bf{Proposition 2}}. The  coordinate systems   $(\\bar U_1 , \\bar V_1) $ and $(\\bar U_2 , \\bar V_2) $ over the quotient space $ERb \/\\hspace{-0.17 cm}\\sim$ are compatible, i.e., the function change of coordinates is a diffeomorphism.\\\\\n\nTo prove it we have to consider the common regions of the domains of $P_1$ and $P_2$, that is $R_1\\cup R_2$. It is enough to prove it in one of them, since in the other one the procedure would be analogue. We choose $R_2$, as shown in Fig. \\ref{entorncopy}. So, we shall choose  $U =U_{+}=  \\frac{1}{2}(\\sqrt{\\varphi^2 +4 \\bar V_2^2} - \\varphi ), \\bar V_2 = V$ given in (\\ref{antecedent+}), and\nwe can express $V\/U$ and $U^2-V^2$ as functions of $(\\bar U_2 , \\bar V_2)$\n\\begin{eqnarray}\n& \\frac{V}{U}= q ( \\bar U_2,\\bar V_2) \\, \\ , \\  q ( \\bar U_2,\\bar V_2)= \\frac{2\\bar V_2}{- \\varphi +\\sqrt{\\varphi^2  + 4 \\bar V_2^2} }\\\\\n& U^2 -V^2 = p ( \\bar U_2,\\bar V_2)\\, \\ , \\  p ( \\bar U_2,\\bar V_2)= \\frac{\\varphi}{2}\\left(\\varphi- \\sqrt{\\varphi^2 + 4\\bar V_2^2 } \\right)\\label{u2mev2} \\ .\n\\end{eqnarray}\nThe functions $p ( \\bar U_2,\\bar V_2)$  and $q ( \\bar U_2,\\bar V_2)$ are both  differentiable and\n moreover $|q ( \\bar U_2,\\bar V_2)| < 1$  because $\\varphi < 0$ in the region considered. This property will be used bellow. Let us introduce $ \\tilde S(\\bar U_2,\\bar V_2)= S\\left(\\sqrt{h_{ER}(p(\\bar U_2,\\bar V_2)}\\right)$, then from equation (\\ref{co1})  we have\n\\begin{eqnarray}\n& \\bar U_1^2 -\\bar V_1^2 = D^2 \\tilde S(\\bar U_2, \\bar V_2)^2\\label{una}\\\\\n&  \\frac{\\bar V_1}{\\bar U_1}= q( \\bar U_2,\\bar V_2)= \\frac{V}{U}\\label{dues}  \\ ,\n\\end{eqnarray}\n and by  combining (\\ref{una}) and (\\ref{dues})  one gets the change of coordinates \n\\begin{equation}\\label{1-2}\n \\bar U_1 =\\frac{D \\tilde{S}(\\bar U_2 , \\bar V_2) }{ \\sqrt{1-q ( \\bar U_2,\\bar V_2)^2}} \\, \\ , \\  \\bar V_1 =\\frac{D  \\tilde{S}(\\bar U_2 , \\bar V_2)  q (\\bar U_2,\\bar V_2)}{\\sqrt{1-q ( \\bar U_2,\\bar V_2)^2}}  \\ .\n\\end{equation}\n The change of coordinates (\\ref{1-2}) is differentiable because $ q^2 < 1$ in the domain considered. This completes the proof. \\\\\n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{canvicoord12.pdf}\n\\caption{Diagram of the change of isothermic  coordinates  $ ( \\bar U_2 , \\bar V_2) \\rightarrow (\\bar U_1 , \\bar V_1)$ in the region $R_2$, that has been proved to be differentiable.}\\label{entorncopy}\n\\end{figure}\n\nWith the  coordinate  systems $ (\\bar U_1 , \\bar V_1) , (\\bar U_2 , \\bar V_2) $ we cover all  the space $\\bar H = \\{(\\bar U , \\bar V ) \\mid  \\bar U^2 - \\bar V^2 > 0 \\, , \\bar U > 0\\} $. \n  {\\bf{Hence  we have given a differentiable structure to the quotient space}} $ERb \/\\hspace{-0.17 cm}\\sim$.  Regarding the geometrical characteristics of this space, the manifold  $ ERb\/ \\hspace{-0.17 cm}\\sim$ \n  is diffeomorphic to the covering space of the hyperboloid of revolution with the point $(D,0)$ removed, as proved in Appendix A. This is  why we have given the name \\textbf{hyperbolic Einstein-Rosen bridge (\\textit{hER})} to the manifold  $ ERb \/ \\hspace{-0.17 cm}\\sim$. Simply, to a point with coordinates $(\\bar U_1, \\bar V_1)$ we associate a point of the hyperboloid with the same coordinates $ (\\bar U , \\bar V)=(\\bar U_1, \\bar V_1)$, and similarly for points of $ERb\/\\hspace{-0.17 cm}\\sim$  with coordinates $(\\bar U_2, \\bar V_2)$. Therefore, the set $\\bar J - (D,0) \\subset \\bar H$, which is the throat of the hyperboloid, is diffeomorphic to the bridge $B \\subset ERb\/ \\hspace{-0.17 cm}\\sim$ that connects the sheets of the Einstein-Rosen space. \\\\\n \n \n \nIn the next section, using the function $P_{2}^{-1}$ introduced in  (\\ref{pedos}) we shall get  the metric  in a neighborhood of the bridge expressed in coordinates $(\\bar U_2 , \\bar V_2)$, and will be able to study the passability of the bridge.\n\n\\section{The  metric on the  hyperbolic Einstein-Rosen bridge $hER$}\\label{mher}\n\n\n   The metric space $(ERb, \\Omega^2_{ER} \\eta) $ with $ \\eta= - dV^2 + dU^2 $, obtained in section \\ref{ERBSec}, is a non maximal extension of the incomplete Einstein-Rosen bridge. In the previous section we have obtained, by identifying points of the bridge, a new manifold $ERb\/\\hspace{-0.17 cm}\\sim$  which is diffeomorphic to the covering space of an hyperboloid of revolution, and we have provided it with a differential structure. We shall explain how to obtain in this new space a maximal extension of the Einstein-Rosen bridge different from the well-known Kruskal-Szequeres black hole. \\\\\n   \nThe dipheomorphism $P_{2}^{-1}: \\bar R_2 \\subset \\bar H_2 \\setminus \\bar J \\rightarrow  R_2 \\subset \\Gamma_2$   will be used now to pullback the metric  on the open set $R_2\\subset\\Gamma_2$ over the open set $\\bar R_2$. \n  So, taking into account (\\ref{antecedent+}), we get\n\\begin{eqnarray}\n& d  U = A d \\bar U_2 + B d  \\bar V_2 \\,  , \\,  d V  = d \\bar V_{2} \\\\\n&  A = \\frac12\\left(1-\\frac{ \\varphi}{\\sqrt{\\varphi^2 + 4\\bar V_2^2}}\\right)\\label{A}\\\\\n&  B= - \\frac{1}{2}\\left(1-\\frac{ \\varphi}{\\sqrt{\\varphi^2 + 4\\bar V_2^2}}\\right)\\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} +\\frac{ 2 \\bar V_2}{\\sqrt{\\varphi^2 + 4 \\bar V_2^2}}\\label{B} \\ ,\n\\end{eqnarray}\nwith  $\\varphi = \\sqrt{\\bar V_2^2+ D^2}-\\bar U_2 $. Using these relations we get the metric in the region $\\bar R_2 \\subset \\bar H_2$, where $\\varphi < 0$,  shown in  Fig.  \\ref{entorncopy1}\n \\begin{eqnarray}\\label{metric2}\n &d\\bar s^2 = \\bar \\Omega^2(\\bar \\eta_{_{\\bar U_2 \\bar U_2}} d \\bar U_2^2+ \\bar \\eta_{_{\\bar V_2 \\bar V_2}} d\\bar V_2^2 +2 \\bar \\eta_{_{\\bar U_2 \\bar V_2}}d \\bar U_2 d \\bar V_2)\n\\\\\n&\\bar \\eta _{_{\\bar V_2 \\bar V_2 }}= B^2 -1 \\, \\ , \\ \\bar \\eta_{_{\\bar U_2 \\bar U_2} }=A^2 \\, \\ , \\, \\bar \\eta_{_{\\bar U_2 \\bar V_2} } = AB \\ ,\n\\end{eqnarray}\nwhere the factor $\\bar \\Omega^2=\\Omega^2_{ER}(U^2-V^2) $ is obtained by substituting $U^2 - V^2 =U_+^2 - V^2= \\frac{\\varphi}{2}\\left(\\varphi- \\sqrt{\\varphi^2 + 4\\bar V_2^2 } \\right)$into the function $\\Omega^2_{ER}$ given in  (\\ref{ERI2}). \n The same functional form of the metric defined above in $\\bar R_2$ {\\bf{is extended}} now to the the region $ (\\bar J -(D,0)) \\cup \\bar R_1$ where $\\varphi \\geq 0$. \nThe extended metric is degenerate over the axis $\\bar V_2=0$ in the region $\\varphi > 0$ (the left hand of the bridge), because there we have $ A=B=0$, but it  is smooth over all the bridge $\\bar J -(D,0)$.\\\\\n\nThe components of the metric are analytic functions in the open set formed by subtracting the semi axis $\\varphi > 0 \\, , \\bar V_2 =0$ from the open set $\\bar H_2$, and so are the components of the Ricci tensor. On the other hand, by construction, the Ricci  tensor is null in the region $\\bar R_2$ (to the right of the bridge), since this region is diffeomorphic to the right hand of the Einstein-Rosen space-time, where the Ricci tensor vanishes. Therefore, according to the theorem of interior uniqueness of analytic functions, the Ricci tensor will also be null in the region $\\bar R_1$ (to the left of the bridge) minus the semi axis $\\varphi > 0 \\, , \\bar V_2 =0$. Then, the metric verifies the Einstein equations in empty space, except in the degenerate region. To ascertain the geometrical character of the degeneracy  we have considered the Kretschmann scalar, $\\mathrm{Kret}(\\bar U_2, \\bar V_2)=R_{abcd}R^{abcd}$ in the region $\\varphi > 0$, $\\bar U_2 < D$, to study the limit when $\\bar V_2$  tends to zero keeping $\\bar U_2$ constant.  We have found that it diverges as $ (D-\\bar U_2)^6\/\\bar V_2^{10} $. The calculation is straightforward and helped by the fact that the metric is the sum of two mutually independent submetrics.\n The semi-axis  $ \\varphi > 0 , \\bar V_2 = 0 $  is therefore a geometrical singularity of the space-time. \\\\\n\n\n\n\n\n\n We shall show that the bridge is a horizon that protects the right region ($\\varphi < 0$) from the singularity present in the left region ($\\varphi > 0$), as in the maximal extension of the Schwarzschild metric.\nWe start  studying the geometrical properties of the hyperbola $\\bar J -(D,0)$.\nA bridge is the possibility of communicating two separate zones. To ascertain if our hyperbola has this property  we shall construct the light cones over it, i.e., the tangent vectors to the pair of light rays at any point of $\\bar J$.\nIf we renounce to an affine parametrization we can  get  them   by considering the simpler metric $ ds^2 =\\bar \\eta_{_{\\bar U_2 \\bar U_2}} d \\bar U_2^2+ \\bar \\eta_{_{\\bar V_2 \\bar V_2}} d\\bar V_2^2 +2 \\bar \\eta_{_{\\bar U_2 \\bar V_2}}d \\bar U_2 d \\bar V_2$.  It is manifest that the transformation $ \\bar V_2 \\rightarrow - \\bar V_2$ is an isometry, therefore the geodesics in the $\\bar V_2 < 0$ region may be obtained by symmetry respect the $\\bar V_2=0$ axis.\nUsing the coordinate $\\bar V_2$ as parameter of the isotropic geodesics, we get an algebraic equation to determine the isotropic directions, which is given by\n\\begin{equation}\n  \\bar \\eta_{_{\\bar U_2 \\bar U_2}}\\left( \\frac{d \\bar U_2}{d \\bar V_2}\\right)^2 + 2 \\eta_{_{\\bar U_2 \\bar V_2}} \\frac{d \\bar U_2}{d \\bar V_2} + \\bar \\eta_{_{\\bar V_2 \\bar V_2}} =0  \\ .\n\\end{equation}\nThe  two solutions  define the following two first order differential equations\n\\begin{equation}\n\\frac{d \\bar U_2}{d \\bar V_2}= \\frac{1-B}{A}\\, \\  \\ ,  \\  \\   \\frac{d \\bar U_2}{d \\bar V_2}=-\\frac{1+B}{A}\\label{isoge12}  \\ ,\n\\end{equation}\nwhere $A$ and $B$ are defined in (\\ref{A}), (\\ref{B}). On the bridge it is verified $\\varphi =0$, so we have \n\\begin{eqnarray}\n&& \\frac{1-B}{A}\\mid_{\\varphi =0} = \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} \\, \\  \\ , \\    -\\frac{1+B}{A}\\mid_{\\varphi =0} =  \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}}-4 \\,  \\     \\mathrm{if}\\,  \\  \\bar V_2 > 0\\\\\n&& \\frac{1-B}{A}\\mid_{\\varphi =0} = \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}}+4 \\, \\  \\ , \\    -\\frac{1+B}{A}\\mid_{\\varphi =0} = \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} \\,  \\     \\mathrm{if}\\,  \\  \\bar V_2 < 0 \\ ,\n \\end{eqnarray}\nand it follows immediately that the parametrization  $( \\bar U_2 = \\sqrt{\\bar V_2^2 + D^2}, \\bar V_2 = \\bar V_2)$ of the  bridge $\\bar J-(D,0)$   is a light ray. One of the two isotropic vectors at any point  $ p \\in \\bar J-(D,0)$ is the tangent vector to the bridge $K_1= (\\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}},1 )$,  while  the other one is clearly directed from right to left:  $K_2 =( -4 + \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} , 1) $ in the $\\bar V_2 > 0 $ region, and from left to right $K_2 =( 4 + \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} , 1) $ in the $\\bar V_2 < 0 $ zone . The couple $(K_1,K_2) $ define the light cone at each point of the bridge.  They show that  in the region $\\bar V_2 > 0$  the matter can traverse the bridge only in right to left direction; by contrary  in the region $\\bar V_2 <  0$  matter only can traverse it from left to right. Therefore we conclude that the bridge is traversable.\\\\\n\nFig. \\ref{Fi:condellum1.pdf} shows the representation of some isotropic geodesics in a neighbourhood of the bridge and the light cones, which show the possible transit directions. These directions agree with the orientation of the  light cone  chosen  at the end of  section \\ref{free}. Let us point out that if we  had started  obtaining the metric on $\\bar R_1\\subset \\bar H_2$ by choosing the value $U = U_{-}$ given in  (\\ref{antecedent-}), instead of $ U = U_{+}$, and extending it to $\\bar R_2$, the bridge would be   passable too but with inverted transit directions. This option would correspond to the opposite choice of the light cone orientation in  section (\\ref{free}).\\\\\n\n\\begin{figure}[hbt] \\label{consher}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{cons3.pdf}\n\\caption{Diagram showing the light rays (red lines) in  a neighbourhood of the bridge using   coordinates $(\\bar U_2, \\bar V_2) $. The central hyperbola (green line), minus $(D,0)$,  is the bridge formed by two interrupted  light rays. The light cones traced on the bridge show the possible directions of transit for light and matter. All  the light  rays have been obtained using  $Mathematica$.}\\label{Fi:condellum1.pdf}. \n\\end{figure}\n\nRegarding the point $(D,0)$, it is a conical singularity and is not included in the space-time. If it had been included in the new manifold, the metric would be discontinuous at that point, and equations (\\ref{isoge12}) would be of the form $\\frac{d\\bar U_2}{d\\bar V_2}= f(\\bar U_2,\\bar V_2)$ with $f(\\bar U_2,\\bar V_2)$ discontinuous at $(D,0)$, no satisfying the existence theorem. \\\\\n\n  An interesting characteristic of  the semi-axis  $( \\varphi > 0 , \\bar V_2 = 0 )$, where the metric is degenerate, is that it cannot be connected by light rays to any point of the space time, as it  is manifest in Fig. \\ref{Fi:condellum1.pdf}. Moreover, light and matter coming from the region $\\varphi > 0 , \\bar V_2 > 0$ cannot traverse to the right side. Therefore, the bridge acts as a horizon and the singularity satisfies the cosmic censorship, since it is causally disconnected from the right side. This singularity could have been avoided by obtaining the metric on the left  and right regions (i.e., $\\bar R_1$ and $\\bar R_2$) independently, as  pull-backs  of the known metrics in regions $ R_1$ and $R_2$. But the metric obtained in this way is not continuous on the bridge, and it cannot be considered an actual extension of the Einstein-Rosen space.\\\\\n\n\\section{The problem of the source of the $hER$ metric space.} \\label{source}\n\n Finally, let us add some comments about  the source of the $hER$ metric space.  According to Katanaev,  a point particle is the  source of the incomplete  metric (\\ref{ER}). However,  the point particle is at $r=0$ in isotropic coordinates, which is at  infinite distance of the bridge\nwhere the curvature of this space-time is enormous. This fact is interpreted by Katanaev as repulsive gravity in the left side of the bridge, \n but for us this is  an uncomfortable conclusion that prompts to look for a different interpretation.\\\\\n \nIn the case of the black-hole extension of the Schwarzschild metric,  an illuminating finding was its relation with the {\\bf{continued gravitational contraction}} of an  spherical star made of pressure-less  matter \\cite{OPENSNY}: only the exterior of the collapsing star was  a part of the Kruskal-Szekeres extension. In our case, it is an interesting  issue  to find the kind of collapse whose exterior would correspond to a part of the hyperbolic  Einstein-Rosen bridge. The spherical star must contract, as shown in Fig. \\ref{colapse}, until the free surface reaches the bridge, and then it must begin to expand again, but now in the other side of it. (Only  negative pressures could produce this sort of collapse. Newtonian gravity predicts a negative gravitational contribution to the pressure, but a correct extension of this gravitational effect has not been accomplished   so far). The orientation of the light cone in Fig. \\ref{colapse} shows that  the source in her expansive phase, in the left part of the bridge, can not be observed from the right of it, and no material particle evolving in the left part can get the bridge.  In other words, an observer in the right side could not distinguish this process from an Oppenheimer-Snyder collapse creating a black-hole. However, after crossing the horizon, the star would not finish in a singularity, but would begin to expand again. In fact, no singularities would be present, since the degenerate region $(\\bar U_2 < D , \\bar V_2=0)$ is now occupied by the collapsing fluid.\\\\\n\nThis interpretation would explain why gravity is stronger  on the bridge, because any point of it is  connected by a light ray (the  bridge) to the collapsing  body at the point of maximum contraction. By contrast, the curvature tends to zero  at great distances to the left of the bridge, because  the light rays connect with points at the expansion phase. \n No point-particle would be present in this space-time, \n  only a part of the mathematical point-particle  Katanaev's  solution  would be  generated as the exterior of this collapse-expansion process.  \\\\\n\n\n\n\n\n\n\n  \n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.35\\textwidth]{con+colapse.pdf}\n\\caption{ Diagram of the proposed collapse process generating a part of the $hER$ space-time. The green line represents the bridge, generated by light rays.   The yellow  region is the source, contracting before and in expansion after passing the bridge.}\\label{colapse}\n\\end{figure}\n\n\\section{Conclusions} \\label{conclusions}\n\n In this work we have obtained a new extension ($hER$) of the original incomplete Einstein-Rosen bridge, different from the well-known Kruskal-Szekeres space-time and from the more recent one obtained by Guendelman et al. \\cite{GUEN1,GUEN2}. The manifold of $hER$ is diffeomorphic to the covering space of an hyperboloid of revolution, and the metric components are analytic functions in a neighborhood of the bridge with a geometrical singularity on the semi axis $\\bar V_2 = 0 , \\bar U_2 < D$. The bridge is formed by light rays, and is traversable unlike the bridge in the $KS$ extension. The $hER$ space-time differs from the one obtained by Guendelman et al. in three aspects: a) it has no source installed in the bridge, b) it has no closed time-like geodesics typical of whormholes with exotic sources, and c) it presents a singularity, though compatible with the cosmic censorship conjecture (it is causally disconnected from the opposite side of the bridge, so the bridge acts as a horitzon). Finally, we have discussed the problem of the source of this space-time, and proposed a gravitational collapse as in the black-hole case but with two phases: contraction in one side and expansion in the other. It would generate a part of the $hER$ metric in the exterior of the collapse, avoiding the singularity. We leave the deep analysis of this process for future work.\\\\\n\n\n{\\it Acknowledgments.--} P. B. is supported by a Ph.D. fellowship, Grant No.  FPU17\/03712. M.P. thanks support by the Spanish \"Ministerio de Economia y Competitividad (sic)\" and the \"Fondo Europeo de Desarrollo Regional\" MINECO-FEDER Project No. PGC2018-095251-B.100.\n\n\\section*{Appendix A: Diffeomorphism with the covering space of the hyperboloid}\\label{HER}\nWe shall obtain the expression in isothermal coordinates of the covering space of a two dimensional hyperboloid with signature $(-,+)$ that has been used in section \\ref{ED}. \nThe manifold  of the  two-dimensional sections with ($\\theta , \\varphi$) constant  of the Einstein-Rosen metric given in (\\ref{ER}) could be considered, excluding the section $u=0$, as  a surface of revolution (for example the one sheet hyperboloid) if we take $t$ as the angle of revolution. This is only possible if we consider the covering space of the hyperboloid, $\\bar H,$ by extending  to all $\\mathbb{R}$ its angular coordinate, defined in $(0, 2\\pi)$.  We shall do that  and   our quotient space  $ERb\/\\hspace{-0.17 cm}\\sim$ defined in section \\ref{ED} will be   \n diffeomorphic to the covering  surface $\\bar H$.\\\\\n \n {\\bf{Description of a one sheet hyperboloid with signature $ ( - \\, \\ +)$}}. We consider in the space  $\\mathbb{R}^3$ the metric $ ds^2 = -dx^2 -dy^2 +dz^2 $, and the surface of an hyperboloid defined  by  $ \\frac{x^2}{q^2} +\\frac{y^2}{q^2}-\\frac{z^2}{p^2}=1$, with $ p > q $. It can be parametrizated as: $X(\\varphi_1 , \\varphi_2)= ( q \\cosh \\varphi_2  \\cos \\varphi_1, q \\cosh \\varphi_2  \\sin \\varphi_1 , p \\sinh  \\varphi_2  ) $,  with $ 0  < \\varphi_1 < 2\\pi \\, , \\  - \\infty < \\varphi_2 < \\infty $, and one obtains  the  metric  $ds^2 = -q^2 \\cosh^2\\varphi_2 \\,d  \\varphi_1^2 + ( p^2 \\cosh^2\\varphi_2 - q^2 \\sinh^2\\varphi_2 ) d \\varphi_2^2$. Now, using  the\n  {\\bf{Proposition 1}} we  can introduce  isothermal coordinates \n  $ \\bar U = f_{{H}}(\\varphi_2) \\cosh (\\varphi_1)$, \n   $ \\bar V = f_{{H}}(\\varphi_2) \\sinh (\\varphi_1)$, \n   with $ f_{H}(\\varphi_2)= D \\exp(\\int_0^{\\varphi_2} \\sqrt{1-\\frac{p^2}{q^2}  \\tanh^2s}\\,  ds )$ and $D$ a  positive arbitrary constant, and  express the  metric as $d\\bar s^2 = q^2 \\cosh^2 (\\varphi_2) f_{H}^{-2}(\\varphi_2) ( -d \\bar V^2 + d\\bar U^2)$.\n The equation $ \\bar U^2 -\\bar V^2 =  f_{H}^2(\\varphi_2)$ defines  a monotonous increasing  function,  $ \\varphi_2 = h_{_{H}}(\\bar U^2-\\bar V^2)$ in the open set  $\\bar U^2-\\bar V^2>0 , \\bar U>0$. It  verifies: $  h_{H}(0)=-\\infty \\, , \\  h_{H}(D^2)= 0 \\, , \\ h_{H}(\\infty)=\\infty $.  In isothermal coordinates, \n the lines $\\varphi_1 = \\mathrm{const}.$ in the $( \\varphi_1 , \\varphi_2)$ plane  transform into the straight lines $ \\bar V = \\tanh(\\varphi_1)\\, \\bar U$ in the $(\\bar U , \\bar V)$ plane, for values $0 < \\varphi_1 < 2 \\pi$; \n  and the circles  $\\varphi_2 = \\mathrm{const}.$ into segments of  the hyperbolae $\\bar U^2 -\\bar V^2 = f_H^2(\\varphi_2)$.  In particular, the circle  $\\varphi_2 = 0$ transforms into a segment of\nthe hiperbola $\\bar J= \\{(\\bar U,\\bar V)\\mid \\bar U^2 -\\bar V^2 = D^2, \\bar U>0 \\}$ in isothermal coordinates. \\\\\n\n  {\\bf{The covering space $\\bar H$ of the  one sheet hyperboloid}}. \nThe covering space of a circle in $ \\mathbb{R}^2$, parametrized as $ X(\\varphi_1,\\varphi_2)=( a \\cos \\varphi_1, a \\sin \\varphi_1)$,  is the helix in $ \\mathbb{R}^3$, parametrized as $X(\\varphi_1,\\varphi_2)=( a \\cos \\varphi_1, a \\sin \\varphi_1 , b \\varphi_1)$. Using this analogy we extend  the coordinate $\\varphi_1$ to obtain the parametrization of the covering space of the one sheet hyperboloid:\n$ X(\\varphi_1 , \\varphi_2)= ( q \\cosh \\varphi_2  \\cos \\varphi_1, q \\cosh \\varphi_2  \\sin \\varphi_1 , p \\sinh  \\varphi_2  , \\  \\varphi_1 )$, with $- \\infty < \\varphi_1 < \\infty\\, , \\  - \\infty < \\varphi_2 < \\infty$.\n The expression in isothermal coordinates of this covering space can be obtained by extending  the range of values of the isothermal coordinates of the hyperboloid to the open set $\\bar H= \\bar U^2 - \\bar V^2 > 0\\, , \\bar U > 0$. \\\\\n\n\n{\\bf{Diffeomorphism $ERb\/\\hspace{-0.17 cm}\\sim\\, \\leftrightarrow \\bar H-(D,0)$}}. We start  with the non isothermal coordinates $ ( u , t ), (\\varphi_2 , \\varphi_1)$, used to express  the Einstein-Rosen metric (\\ref{ER}) and the covering space of the one sheet hyperboloid respectively. Let us consider the following diffeomorphisms: $\\varphi_2 =  u\\,,\\,\\varphi_1 = \\frac{t}{4m}$, for the open $u>0$ and $\\varphi_2 =  u\\,,\\,\\varphi_1 = -\\frac{t}{4m}$ for the open $u<0$. Now, by considering the inverse change $ u=\\mathrm{sig} (U)\\sqrt{h_{ER}(U^2-V^2)}$ (see section \\ref{ERBSec}) and the well-known  hyperbolic trigonometric relations, we  get the expressions   of this assignations in isothermal coordinate\n\\begin{eqnarray}\n&  \\bar U =  f_H\\left(\\mathrm{sig}(U)\\sqrt{h_{ER}(U^2-V^2)}\\right)\\frac{\\mid U \\mid}{\\sqrt{U^2-V^2}}\\label{barU1}\\,\\,\\,\\\\\n&  \\bar V =  f_H\\left(\\mathrm{sig}(U)\\sqrt{h_{ER}(U^2-V^2)}\\right)\\frac{V}{\\sqrt{U^2-V^2}} \\ .\\label{barV1}\n\\end{eqnarray}\nNote that they coincide with the expressions of the map $P_1$, given by \\eqref{co1}, if we define $S= f_{H}\/D$. If we had chosen a different surface of revolution diffeomorphic to the hyperboloid, we would had obtained a different function $S(\\cdot)$ with similar characteristics. That is why in the definition of $P_1$ we have generalized the function $S(\\cdot)$ to any positive, smooth and monotonous increasing  function verifying $S(0)=1$. Therefore, any point of the manifold $ERb\/\\hspace{-0.17 cm}\\sim$ defined with the coordinates $(\\bar U_1,\\bar V_1)$ (all points except the bridge), can be related with the hyperboloid (or a similar surface of revolution) by the diffeomorphism $\\bar U_1=\\bar U\\,,\\,\\bar V_1=\\bar V$, covering all the surface except the throat $\\bar J$. The points of the bridge (and its neighbourhood) in the manifold $ERb\/\\hspace{-0.17 cm}\\sim$ can be expressed with the coordinates $(\\bar U_2,\\bar V_2)$, defined in \\eqref{U2V22}. The bridge correspond to the set $B =\\{(\\bar U_2,\\bar V_2)\\mid  \\bar U_2^2 - \\bar V_2^2= D^2\\, , \\bar U_2 > 0\\}-(D,0)$, so it can be related to the throat of the hyperboloid by the simple diffeomorphism $\\bar U_2=\\bar U\\,,\\,\\bar V_2=\\bar V$. In the Proposition 2 we have proved that the coordinate systems $(\\bar U_1,\\bar V_1)$ and $(\\bar U_2,\\bar V_2)$ are compatible in the intersection of their domains, so there are no discontinuities. Therefore the manifold $ERb\/\\hspace{-0.17 cm}\\sim$ is diffeomorphic to the covering space of the one sheet hyperboloid $\\bar H$, after removing the point $(D,0)$.\\\\\n\n\\section*{Appendix B: Definitions} \\label{definitions}\n\nWe shall summarize the main definitions used in the paper in order to facilitate the reading.\\\\\n\n$ ERb = \\{(U,V) \\in \\mathbb{R}^2 \\mid U^2-V^2 \\geq 0\\} - (0,0)$ \\\\\n \n$ \\partial  ERb =  \\{(U,V) \\in \\mathbb{R}^2 \\mid U^2-V^2 =0\\} - (0,0)$ \\\\\n\n $R_1\\subset ERb =  \\{(U,V) \\in \\mathbb{R}^2 \\mid 0 < U^2-V^2 < \\epsilon^2, U < 0 \\} \\}$ \\\\\n \n $R_2 \\subset ERb =  \\{(U,V) \\in \\mathbb{R}^2 \\mid 0 < U^2-V^2 < \\epsilon^2, U > 0 \\} \\}$ \\\\\n \n $ \\Gamma_1 \\subset ERb \\, , \\, \\Gamma_1 =  \\{(U,V) \\in \\mathbb{R}^2 \\mid 0 < U^2-V^2 \\} $\\\\\n \n  $ \\Gamma_2 \\subset ERb \\, , \\, \\Gamma_2 = \\{(U,V) \\in \\mathbb{R}^2 \\mid 0 \\le U^2-V^2 < \\epsilon^2 \\} - (0,0)$ \\\\\n\n \n $\\bar H = \\{(\\bar U , \\bar V )\\in \\mathbb{R}^2  \\mid  \\bar U^2 - \\bar V^2 > 0 \\, , \\bar U > 0\\} $ \\\\\n\n$ \\bar J \\subset \\bar H = \\{(\\bar U , \\bar V )\\in \\mathbb{R}^2  \\mid  \\bar U^2 - \\bar V^2 = D^2 \\, , \\bar U > 0\\} $ \\\\\n \n $ \\bar H_1 \\subset \\bar H \\, , \\, \\bar H_1 = \\{(\\bar U_1, \\bar V_1 )\\in \\mathbb{R}^2  \\mid   \\bar U_1^2 - \\bar V_1^2 > 0 \\, , \\bar U_1 > 0\\} \\setminus\\bar J$ \\\\\n \n    $ \\bar H_2 \\subset \\bar H \\, , \\ \\bar H_2 = \\{(\\bar U_2, \\bar V_2 )\\in \\mathbb{R}^2  \\mid \\bar U_L(\\bar V_2) <  \\bar U_2 < \\bar U_R(\\bar V_2)\\}- (D,0) $\\\\\n    \n $ P_1 : \\Gamma_1\\subset ERb \\rightarrow \\bar H_1 \\subset \\bar H $\\\\\n \n  $ P_2 : \\Gamma_2\\subset ERb \\rightarrow  \\bar H_2 \\subset \\bar H $\\\\\n  \n  \n  $\\bar R_1 =\\{(\\bar U_2, \\bar V_2 )\\in (\\bar H_2 - \\bar J ) \\mid  \\bar U_2 < \\sqrt{D^2 + \\bar V_2^2}\\}$\\\\\n  \n  $\\bar R_2 =\\{(\\bar U_2, \\bar V_2 )\\in (\\bar H_2 - \\bar J ) \\mid  \\bar U_2 > \\sqrt{D^2 + \\bar V_2^2}\\}$\\\\\n  \n  \n    $P_2(R_i) = \\bar R_i \\, , \\ i = 1, 2$\\\\\n    \n\n\n$\\varphi = \\sqrt{\\bar V_2^2 + D^2} - \\bar U_2 $\\\\\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\nA {\\it dominating set} in a graph $G$ is a subset \nof vertices $S$ such that every vertex in $V(G)\\setminus S$\nis a neighbour of some vertex of $S$. The {\\it domination number} \nof $G$ is the minimum size of a dominating set of $G$. We \ndenote it by $\\gamma(G)$. This paper is devoted to the calculation \nof the domination number of complete grids.\n\nThe notation $[i]$ denotes the set $\\{1,2,\\ldots, i\\}$. If $w$ is a\nword on the alphabet $A$, $w[i]$ is the $i$-th letter of $w$, and \nfor every $a$ in  $A$, $\\vert w \\vert_a$ denotes the number of occurrences of \n$a$ in $w$ (i.e. $\\vert \\{ i\\in \\{1,\\ldots,\\vert w\\vert\\} \n: w[i]=a\\}\\vert$). \nFor a vertex $v$, $N[v]$ denotes the closed neighbourhood of $v$ \n(i.e. the set of\nneighbours of $v$ and $v$ itself). For a subset of\nvertices $S$ of a vertex set $V$ of a graph, we denote by $N[S] =\n\\bigcup_{v\\in S}N[v]$.  Note that $D$ is a dominating set of $G$ if\nand only if $N[D] = V(G)$. Let $G_{n,m}$ be the $n\\times m$ complete\ngrid, i.e. the vertex set of $G_{n,m}$ is $V_{n,m}:=[n]\\times [m]$, and two\nvertices $(i,j)$ and $(k,l)$ are adjacent if\n$|k-i|+|l-j|=1$. The couple $(1,1)$ denotes the bottom-left vertex of\nthe grid and the couple $(i,j)$ denotes the vertex of the $i$-th column\nand the $j$-th row. We will always assume in this paper that $n\\leq\nm$. \nLet us illustrate our purpose by an example of a dominating set of the\ncomplete grid $\\fG{24}{24}$ on Figure~\\ref{fig:24x24}.\n\n\\begin{figure}\n  \\centering\n  \\psset{unit=0.28cm}\n  \\include{d24x24}\n  \\caption{Example of a set of size 131 dominating the grid $\\fG{24}{24}$\\label{fig:24x24}}\n\\end{figure}\n\nThe first results on the domination number of grids were obtained\nabout 30 years ago with the exact values of $\\gG2n$, $\\gG3n$, and $\\gG4n$\nfound by Jacobson and Kinch \\cite{JK} in 1983. In 1993,  \nChang and Clark~\\cite{CC} found those of $\\gG5n$\nand $\\gG6n$. These results were obtained analytically.\nChang~\\cite{Chang} devoted his PhD thesis to study the domination\nnumber of grids; he conjectured that this invariant\nbehaves well provided that $n$ is large enough.\nSpecifically, Chang conjectured the following:\n\n\\begin{conjecture}[\\cite{Chang}]\\label{conj}\n  For every $16\\le n \\le m$, $$\\gamma(G_{n,m}) = \\form{n}{m}.$$\n\\end{conjecture}\n\nObserve that for instance, this formula would give 131 for the domination number \nof the grid in Figure~\\ref{fig:24x24}. To motivate his bound, Chang proposed some constructions of\ndominating sets achieving the upper bound:\n\n\\begin{lemma}[\\cite{Chang}]\\label{lem:cgang}\n  For every $8\\le n\\le m$, $$\\gamma(G_{n,m}) \\le \\form{n}{m}$$\n\\end{lemma}\n\nLater, some algorithms based on dynamic programming were designed to\ncompute a lower bound of $\\gG{n}{m}$. There were numerous\nintermediate results found for $\\gG{n}{m}$ for small values of\n$n$ and $m$ (see \\cite{CCH,HHH,SP} for details). In 1995, Hare,\nHedetniemi and Hare \\cite{HHH} gave a polynomial time algorithm to\ncompute $\\gamma(G_{n,m})$ when $n$ is fixed. Nevertheless, this\nalgorithm is not usable in practice when $n$ hangs over $20$. Fisher\n\\cite{Fisher} developed the idea of searching for periodicity in the\ndynamic programming algorithms and using this technique, he found the\nexact values of $\\gG{n}{m}$ for all $n\\leq 21$. We recall these values\nfor the sake of completeness.\n\n\\begin{theorem}[\\cite{Fisher}]\n  For all $n \\le m$ and $n \\le 21$, we have:\n  $$\\gG{n}{m} = \\left\\{\n    \\begin{array}{ll}\n      \\frceil{m}{3} & \\ \\mbox{\\rm{if}} \\  n=1 \\\\*[0.1cm]\n      \\frceil{m+1}{2} & \\ \\mbox{\\rm{if}} \\  n=2 \\\\*[0.1cm]\n      \\frceil{3m+1}{4} & \\ \\mbox{\\rm{if}} \\  n=3 \\\\*[0.1cm]\n      m+1 & \\ \\mbox{\\rm{if}} \\  n=4  \\ \\mbox{\\rm{and}} \\  m=5,6,9\\\\*[0.1cm]\n      m & \\ \\mbox{\\rm{if}} \\  n=4  \\ \\mbox{\\rm{and}} \\  m\\neq 5,6,9\\\\*[0.1cm]\n      \\frceil{6m+4}{5} - 1 & \\ \\mbox{\\rm{if}} \\  n=5 \\ \\mbox{\\rm{and}} \\  m=7 \\\\*[0.1cm]\n      \\frceil{6m+4}{5} & \\ \\mbox{\\rm{if}} \\  n=5 \\ \\mbox{\\rm{and}} \\  m\\neq 7 \\\\*[0.1cm]\n      \\frceil{10m+4}{7} & \\ \\mbox{\\rm{if}} \\  n=6 \\\\*[0.1cm]\n      \\frceil{5m+1}{3} & \\ \\mbox{\\rm{if}} \\  n=7 \\\\*[0.1cm]\n      \\frceil{15m+7}{8} & \\ \\mbox{\\rm{if}} \\  n=8 \\\\*[0.1cm]\n      \\frceil{23m+10}{11} & \\ \\mbox{\\rm{if}} \\  n=9 \\\\*[0.1cm]\n      \\frceil{30m+15}{13} - 1 & \\ \\mbox{\\rm{if}} \\  n=10 \\ \\mbox{\\rm{and}} \\  m\\equiv_{13}10 \\ \\mbox{\\rm{or}} \\  m=13,16\n      \\\\*[0.1cm]\n      \\frceil{30m+15}{13} & \\ \\mbox{\\rm{if}} \\  n=10 \\ \\mbox{\\rm{and}} \\  m\\not\\equiv_{13}10 \\ \\mbox{\\rm{and}} \\  m\\neq13,16 \\\\*[0.1cm]\n      \\frceil{38m+22}{15} - 1 & \\ \\mbox{\\rm{if}} \\  n=11 \\ \\mbox{\\rm{and}} \\  m=11,18,20,22,33 \\\\*[0.1cm]\n      \\frceil{38m+22}{15} & \\ \\mbox{\\rm{if}} \\  n=11 \\ \\mbox{\\rm{and}} \\  m\\neq11,18,20,22,33 \\\\*[0.1cm]\n      \\frceil{80m+38}{29} & \\ \\mbox{\\rm{if}} \\  n=12 \\\\*[0.1cm]\n      \\frceil{98m+54}{33} - 1 & \\ \\mbox{\\rm{if}} \\  n=13 \\ \\mbox{\\rm{and}} \\  m\\equiv_{33}13,16,18,19 \\\\*[0.1cm]\n      \\frceil{98m+54}{33} & \\ \\mbox{\\rm{if}} \\  n=13 \\ \\mbox{\\rm{and}} \\  m\\not\\equiv_{33}13,16,18,19 \\\\*[0.1cm]\n      \\frceil{35m+20}{11} - 1 & \\ \\mbox{\\rm{if}} \\  n=14 \\ \\mbox{\\rm{and}} \\  m\\equiv_{22}7\\\\*[0.1cm]\n      \\frceil{35m+20}{11} & \\ \\mbox{\\rm{if}} \\  n=14 \\ \\mbox{\\rm{and}} \\  m\\not\\equiv_{22}7\\\\*[0.1cm]\n      \\frceil{44m+28}{13} - 1 & \\ \\mbox{\\rm{if}} \\  n=15 \\ \\mbox{\\rm{and}} \\   m\\equiv_{26}5\\\\*[0.1cm]\n      \\frceil{44m+28}{13} & \\ \\mbox{\\rm{if}} \\  n=15 \\ \\mbox{\\rm{and}} \\   m\\not\\equiv_{26}5\\\\*[0.1cm]\n      \\frfloor{(n+2)(m+2)}{5}-4 & \\ \\mbox{\\rm{if}} \\  n\\ge16 \\\\*[0.1cm]\n    \\end{array}\\right.$$\n  \n\\end{theorem}\n\nNote that these values are obtained by specific formulas for every $1\\leq n\\leq 15$ and by\nthe formula of Conjecture~\\ref{conj} for every $16\\le n \\le 21$. This\nproves Chang's conjecture for all values $16\\le n\\le 21$.\n\nIn 2004, Conjecture~\\ref{conj} has been confirmed up to an \nadditive constant:\n\n\\begin{theorem}[Guichard \\cite{Guichard}] For every $16\\le n \\le m$,\n  $$\\gamma(G_{n,m}) \\geq \\left \\lfloor \\frac{(n+2)(m+2)}{5} \\right\n  \\rfloor -9.$$\n\\end{theorem}\n\nIn this paper, we prove Chang's conjecture, hence finishing the\ncomputation of $\\gG{n}{m}$.  We adapt Guichard's ideas to improve the\nadditive constant from $-9$ to $-4$ when $24 \\le n \\le m$.  Cases\n$n=22$ and $n=23$ can be proved in a couple of hours using Fisher's\nmethod (described in~\\cite{Fisher}) on a modern computer.  They can be\nalso proved by a slight improvement of the technique presented in the\nnext section.\n\n\\section{Values of $\\gamma(G_{n,m})$ when $24 \\le n \\le m$}\n\nOur method follows the idea of Guichard \\cite{Guichard}. A slight\nimprovement is enough to give the exact bound.  \n\nA vertex of the grid $G_{n,m}$ dominates at most 5 vertices (its four\nneighbours and itself). It is then clear that $\\gG{n}{m} \\ge\n\\frac{n\\times m}{5}$. The previous inequality would become an equality\nif there would be a dominating set $D$ such that every vertex of\n$G_{n,m}$ is dominated only once, and all vertices of $D$ are of degree 4\n(i.e.  dominates exactly 5 vertices); in this case, we would have\n$5\\times|D| - n\\times m= 0$. This is clearly impossible (e.g. to\ndominate the corners of the grid, we need vertices of degree at most\n3). Therefore, our goal is to find a dominating set $D$ of $G_{n,m}$ such\nthat the difference $5\\times |D| - n\\times m$ is the smallest.\n\nLet $S$ be a subset of $V(G_{n,m})$. The \\emph{loss} of $S$ is $\\los{S}=5\n\\times \\vert S \\vert - \\vert N[S] \\vert$.\n\n\\begin{proposition} \\label{props}\n  The following properties of the loss function are straightforward:\n  \\begin{enumerate}[(i)]\n  \\item For every $S$, $\\los{S} \\ge 0$ (positivity), \\label{props:1}\n  \\item If $S_1\\cap S_2=\\emptyset$, then $\\los{S_1\\cup S_2} =\n    \\los{S_1} + \\los{S_2} + \\vert N[S_1] \\cap N[S_2] \\vert$, \\label{props:2}\n  \\item If $S'\\subseteq S$, then $\\los{S'}\\le \\los{S}$ (increasing\n    function), \\label{props:3}\n  \\item If $S_1\\cap S_2=\\emptyset$, then $\\los{S_1\\cup S_2} \\ge\n    \\los{S_1} + \\los{S_2}$ (super-additivity). \\label{props:4}\n  \\end{enumerate}\n\\end{proposition}\n\nLet us denote by $\\ell_{n,m}$ the minimum of $\\los{D}$ when $D$ is \na dominating set of $G_{n,m}$. \n\n\\begin{lemma}\\label{lemloss}\n  $\\gamma(G_{n,m}) = \\left \\lceil \\frac{n\\times m + \\ell_{n,m}}{5} \\right\n  \\rceil$\n\\end{lemma}\n\n\\begin{proof}\n  If $D$ is a dominating set of $G_{n,m}$, then $\\los{D} = 5 \\times \\vert\n  D \\vert - \\vert N[D]\\vert = 5 \\times \\vert\n  D \\vert - n\\times m$. Hence, by minimality of $\\ell_{n,m}$ and\n  $\\gamma(G_{n,m})$, we have $\\ell_{n,m}= 5\\times \\gamma(G_{n,m}) - n\\times m$.\n\\end{proof}\n\nOur aim is to get a lower bound for $\\ell_{n,m}$. As the reader can observe\nin Figure~\\ref{fig:24x24}, the loss is concentrated on the border of\nthe grid. We now analyse more carefully the loss generated by the\nborder of thickness $10$. \n\n\\begin{figure}\n  \\centering\n  \\psset{unit=0.28cm}\n  \\begin{minipage}{0.9\\linewidth}\n    \\centering \\include{B}\n  \\end{minipage}\n  \\caption{The graph $\\fG{30}{40}$. The set $\\fBp{30}{40}$ is the set\n    of vertices filled in black. The set $\\fB{30}{40}$ is the set of\n    vertices filled in black or in gray.\\label{fig:bnm}}\n\\end{figure}\n\nWe define the border $B_{n,m} \\subseteq V_{n,m}$ of $G_{n,m}$ as the set of\nvertices $(i,j)$ where $i\\le 10$, or $j\\le 10$, or $i>n-10$, or\n$j>m-10$ to which we add the four vertices\n$(11,11),(11,m-10),(n-10,11),(n-10,m-10)$.  Given a subset $S\\subseteq\nV$, let $I(S)$ be the \\emph{internal vertices} of $S$, i.e. $I(S)=\n\\{v\\in S : N[v]\\subseteq S\\}$.  These sets are illustrated in\nFigure~\\ref{fig:bnm}.  We will compute $b_{n,m} = \\min_D \\los{D}$, where\n$D$ is a subset of $B_{n,m}$ and dominates $I(B_{n,m})$, i.e. $D\\subseteq\nB_{n,m}$ and $I(B_{n,m}) \\subseteq N[D]$. Observe that this lower bound $b_{n,m}$\nis a lower bound of $\\ell_{n,m}$. Indeed, for every dominating set $D$ of\n$G_{n,m}$, the set $D':=D\\cap B_{n,m}$ dominates $I(B_{n,m})$, hence $b_{n,m}\\leq\n\\los{D'}\\leq \\los{D}$. In the remainder, we will compute $b_{n,m}$ and we\nwill show that $b_{n,m} = \\ell_{n,m}$.\n\n\n\\bigskip\n\nIn the following, we split the border $B_{n,m}$ in four parts, $O_{m-12},\nP_{n-12}, Q_{m-12}, R_{n-12}$, which are defined just below.\n\n\\begin{figure}\n  \\centering\n  \\psset{unit=0.28cm}\n  \\include{Q}\n  \\caption{The set $\\fP{19}$ (black and gray), the set of input vertices\n    (gray circles) and the set of output vertices (gray squares).\\label{fig:ppcpa}}\n\\end{figure}\n\nFor $p\\ge 12$, let $P_p\\subset B_{n,m}$ defined as follows : $P_p = ([10]\n\\times \\{12\\}) \\cup ([11] \\times \\{11\\} ) \\cup ([p] \\times [10])$. We\ndefine the \\emph{input vertices} of $P_p$ as $[10]\\times\\{12\\}$ and\nthe \\emph{output vertices} of $P_p$ as $\\{p\\} \\times [10]$. The set\n$P_p$, illustrated for $p=19$ in Figure~\\ref{fig:ppcpa}, corresponds\nto the set of black and gray vertices. The input vertices are the gray\ncircles, and the output vertices are the gray squares. Recall that in\nour drawing conventions, the vertex $(1,1)$ is the bottom-left vertex\nand hence the vertex $(i,j)$ is in the $i^{th}$ column from the left\nand in the $j^{th}$ row from the bottom.\n\n\\begin{figure}\n  \\centering\n  \\psset{unit=0.28cm}\n  \\begin{minipage}{0.9\\linewidth}\n    \\centering \\include{Z}\n  \\end{minipage}\n  \\caption{The sets $O_{m-12}$, $P_{n-12}$, $Q_{m-12}$ and $R_{n-12}$.\n    \\label{fig:pqrs}}\n\\end{figure}\n\nFor $n,m\\in \\mathbb{N}^*$, let $f_{n,m}: [n]\\times[m] \\to [m]\\times[n]$ be\nthe bijection such that $f_{n,m}(i,j) = (j,n-i+1)$. It is clear that the\nset $B_{n,m}$ is the disjoint union of the following four sets depicted\nin Figure~\\ref{fig:pqrs}: $\\fP{n-12}$, $\\fQ{m-12}=f_{n,m}(\\fP{m-12})$,\n$\\fR{n-12} = f_{m,n} \\circ f_{n,m} (\\fP{n-12})$ and $\\fS{m-12}= f_{n,m}^{-1}(\\fP{m-12})$. Similarly to $\\fP{n-12}$, the sets\n$O_{m-12}$, $Q_{n-12}$ and $R_{m-12}$ have input and output\nvertices. For instance, the output vertices of $Q_{m-12}$ correspond\nin Figure~\\ref{fig:ppcpa} to the white squares. Every set playing a\nsymmetric role, we now focus on $P_{n-12}$.\n\n\\bigskip\n\nGiven a subset $S$ of $V_{n,m}$, let the labelling $\\phi_S : V_{n,m} \\to\n\\{0,1,2\\}$ be such that \n\n$$\\phi_S(i,j) = \\left\\{ \n  \\begin{array}{l}\n    0 \\quad \\mbox{if $(i,j)\\in S$} \\\\\n    1 \\quad \\mbox{if $(i,j)\\in N[S]\\setminus S$} \\\\\n    2 \\quad \\mbox{otherwise}\n  \\end{array}\\right.$$\n\nNote that $\\phi_S$ is such that any two\nadjacent vertices in $G_{n,m}$ cannot be labelled~$0$ and~$2$.\n\nGiven $p\\ge 12$ and a set $S\\subseteq P_p$, the \\emph{input word} \n(resp. \\emph{output word}) of $S$ for $P_p$, denoted by $w^{in}(S)$\n(resp. $w^{out}_p(S)$), is the ten letters word on the alphabet\n$\\{0,1,2\\}$ obtained by reading $\\phi_{S}$ on the input vertices \n(resp. output vertices) of $P_p$.\nMore precisely, its $i^{th}$ letter is $\\phi_{S}(i,12)$ (resp. $\\phi_{S}(p,i)$). \nSimilarly, $O_p$, $Q_p$ and $R_p$ have also input and output words. \nFor example, the output word of $S\\subseteq O_p$ for $O_p$ is $w^{out}_p(f_{n,m}(S))$.\n\nAccording to the definition of $\\phi$, the input and output words\nbelong to the set $\\mathcal{W}$ of ten letters words on $\\{0,1,2\\}$ which\navoid $02$ and $20$. The number of $k$-digits trinary numbers without\n$02$ or $20$ is given by the following formula~\\cite{Fisher}:\n\\begin{equation}\n  \\frac{(1+\\sqrt{2})^{k+1}+(1-\\sqrt{2})^{k+1}}{2}\\label{eqn:trinary_words}\n\\end{equation}\nThe size of $\\mathcal{W}$ is therefore $|\\mathcal{W}| = 8119$.\n\n\nGiven two words $w,w' \\in \\mathcal{W}$, we define\n$\\mD^{w,w'}_p$ as the family of subsets $D$ of $\\fP{p}$ such that:\n\\begin{itemize}\n\\item $D$ dominates $I(P_p)$,\n\\item $w$ is the input word $w^{in}(D)$,\n\\item $w'$ is the output word $w^{out}_p(D)$.\n\\end{itemize}\n\nA relevant information for our calculation will be to know, for two\ngiven words $w,w'\\in \\mathcal{W}$, the minimum loss over all losses $\\los{D}$\nwhere $D\\in \\mD^{w,w'}_p$.  For this purpose, we introduce the $8119\\times\n8119$ square matrix $C_p$. For $w,w'\\in \\mathcal{W}$, let\n$C_p[w,w']=\\min_{D\\in\\mD^{w,w'}_p} \\los{D}$ where the minimum of the\nempty set is $+\\infty$.\n\nLet $w,w'\\in \\mathcal{W}$ be two given words. Due to the symmetry of $P_{12}$\nwith respect to the first diagonal (bottom-left to top-right) of the\ngrid, if a vertex set $D$ belongs to $\\Dww{12}$, then $D' = \\{(j,i) |\n(i,j)\\in D\\}$ belongs to $\\mathcal{D}^{w',w}_{12}$. Moreover, it is clear\nthat, always due to the symmetry, $\\los{D}=\\los{D'}$. Therefore, we\nhave $C_{12}[w,w'] = C_{12}[w',w]$ and thus $C_{12}$ is a\nsymmetric matrix.  Despite the size of $C_{12}$ and the size of\n$P_{12}$ (141 vertices), it is possible to compute $C_{12}$ in\nless than one hour by computer. Indeed, we choose a sequence of subsets\n$X_0=\\emptyset, X_1, \\ldots, X_{141}=P_{12}$ such that for every $i\\in\n\\{1,\\ldots, 141\\}$, $X_i\\subseteq X_{i+1}$ and $X_{i+1} \\setminus X_i\n= \\{ x_{i+1} \\}$.  Moreover, we choose the sequence such that for\nevery $i$, $X_i \\setminus I(X_i)$ is at most $21$. This can be done\nfor example by taking $x_{i+1}= \\min \\{ (x,y) : (x,y)\\in P_{12}\n\\setminus X_i \\}$, where the order is the lexical order. For $i\\in\n\\{0,\\ldots, 141\\}$, we compute for every labeling $f \\in \\mathcal{F}_i$, where\n$\\mathcal{F}_i$ is the set of functions $(X_i\\setminus I(X_i)) \\to \\{0,1,2\\}$,\nthe minimal loss $l_{i,f}$ of a set $S\\subseteq X_i$ which dominates\n$I(X_i)$ and such that $\\phi_S(v)=f(v)$ for every $v\\in X_i \\setminus\nI(X_i)$. Note that not every labeling is possible since two adjacent\nvertices cannot be labeled~$0$ and~$2$. The number of possible\nlabellings can be computed using formula~(\\ref{eqn:trinary_words}),\nand since $|X_i \\setminus I(X_i)|$ can be covered by a path of at most\n$23$ vertices, this gives, in the worst case, that this number is less\nthan $10^9$ and can be then processed\nby a computer. We compute inductively the sequence $(l_{i,f})_{f\\in\n  \\mathcal{F}_i}$ from the sequence $(l_{i-1,f})_{f\\in \\mathcal{F}_{i-1}}$ by dynamical\nprogramming, and $C$ is easily deduced from $(l_{141,f})_{f\\in\n  \\mathcal{F}_{141}}$.\n\n\\bigskip\n\nIn the following, our aim is to glue $P_{n-12},\nQ_{m-12},R_{n-12},$ and $O_{m-12}$ together. For two consecutive parts\nof the border, say $P_{n-12}$ and $Q_{m-12}$, the output word of\n$Q_{m-12}$ should be compatible with the input word of $P_{n-12}$.\nTwo words $w,w'$ of $\\mathcal{W}$ are \\emph{compatible} if the sum of their\ncorresponding letters is at most 2, i.e. $w[i]+w'[i]\\leq 2$ for all\n$i\\in [9]$. Note that $w[10] + w'[10]$ should be greater than 2 since\nthe corresponding vertices can be dominated by some vertices of $V_{n,m}\n\\setminus B_{n,m}$.\n\nGiven two words $w,w'\\in \\mathcal{W}$, let\n$\\ell(w,w')= \\vert \\{ i\\in [10] : w[i]\\ne 2 \\text{ and } w'[i]=0 \\}\n\\vert + \\vert \\{ i\\in [10] : w'[i]\\ne 2 \\text{ and } w[i]=0 \\} \\vert$.\n\n\\begin{lemma} \\label{lem:compatible}Let $D$ be a dominating set of $G_{n,m}$ and let us denote\n  $D_P = D\\cap P_{n-12}$ and $D_Q = D\\cap Q_{m-12}$. Then $\\ell(D\\cap\n  (P_{n-12} \\cup Q_{m-12}))= \\ell(D_P) + \\ell(D_Q) + \\ell(w,w')$, where\n  $w=w^{in}(D_P)$ and $w'=w^{out}_q(f_{n,m}^{-1}(D_Q))$. Moreover, $w$ and $w'$\n  are compatible.\n\\end{lemma}\n\n\\begin{proof}\n  By Proposition \\ref{props}(\\ref{props:2}), $\\ell(D\\cap (P_{n-12} \\cup Q_{m-12}))=\n  \\ell(D_P) + \\ell(D_Q) + \\vert N[D_P] \\cap N[D_Q]\\vert$. It suffice then to note that\n  $\\ell(w,w')= \\vert N[D_P] \\cap N[D_Q]\\vert$ to get $\\ell(D\\cap (P_{n-12} \\cup Q_{m-12}))=\n  \\ell(D_P) + \\ell(D_Q) + \\ell(w,w')$.\n  \n  In what remains, we prove that $w$ and $w'$ are compatible.  If\n  those two words were not compatible, there would exist an index $i\\in[9]$\n  such that $w^{out}_{m-12}(f_{n,m}^{-1}(D_Q))[i] + w^{in}(D_P)[i] > 2$.  Thus\n  at least one of these two letters should be a 2, and the other one\n  should not be 0.\n  \n  Suppose that $w^{out}_{m-12}(f_{n,m}^{-1}(D_Q))[i] = 2$ and note that\n  this means that the vertex $(i,13)$ is not dominated by a vertex in\n  $D_Q$. Since $D$ is a dominating set of $G_{n,m}$, every output vertex of\n  $Q_{m-12}$ except $(10,13)$ (and every input vertex of $P_{n-12}$\n  except $(10,12)$) is dominated by a vertex of $D_Q$ or by a vertex of\n  $D_P$. Thus $(i,13)$ should be dominated by its unique neighbour in\n  $P_{n-12}$, $(i,12)$. This would imply that $(i,12)\\in D$\n  contradicting the fact that $w^{in}(D_P)[i] \\neq 0$.\n\n  Similarly, if $w^{in}(D_P)[i] =2$, the vertex $(i,12)$ \n  is not dominated by a vertex in $D_P$, thus $(i,12)$ must be dominated \n  by the vertex $(i,13)\\in D$, contradicting the fact that\n  $w^{out}_{m-12}(f_{n,m}^{-1}(D_Q))[i] \\neq 0$.\n\\end{proof}\n\nLemma~\\ref{lem:compatible} is designed for the two consecutive parts\n$P_{n-12}$ and $Q_{m-12}$ of the border of $G_{n,m}$. Its easy to see\nthat this extends to any pair of consecutive parts of the\nborder, i.e. $Q_{m-12}$ and $R_{n-12}$, $R_{n-12}$ and $O_{m-12}$,\n$O_{m-12}$ and $P_{n-12}$.\n\n\\bigskip\n\nWe define the matrix $8119\\times 8119$ square matrix $L$ which\ncontains, for every pair of words $w,w'\\in \\mathcal{W}$, the value $\\ell(w,w')$:   \n$$L[w,w']=\n\\begin{cases} \n  +\\infty \\text{ if $w$ and $w'$ are not compatible,}\\\\\n  \\ell(w,w') \\text{ otherwise.}\n\\end{cases}$$\nNote that $L$ is symmetric since $\\ell(w,w') = \\ell(w',w)$.\n\nLet $\\otimes$ be the matrix multiplication in $(\\min,+)$ algebra, \ni.e. $C = A \\otimes B$ is the matrix where for all $i,j$, $C[i,j] =\n\\min_k A[i,k] + B[k,j]$. \n\nLet $M_p=L \\otimes C_p$ for $p\\ge 12$. \n\nBy construction, $M_{n-12}[w,w']$ corresponds to the minimum\npossible loss $\\los{D\\cap P_{n-12}}$ of a dominating set $D\\subseteq\nV_{n,m}$ that dominates $I(P_{n-12})$ and such that $w$ is the output\nword of $Q_{m-12}$ and $w'$ is the output word of $P_{n-12}$.\n\n\\begin{lemma}\n  \\label{lem:bnm}\n  For all $24 \\le n \\le m$, we have \n  $$b_{n,m}\\geq \\min_{w_1,w_2,w_3,w_4\\in \\mathcal{W}} M_{n-12}[w_1,w_2] +\n  M_{m-12}[w_2,w_3] + M_{n-12}[w_3,w_4]+ M_{m-12}[w_4,w_1].$$\n\\end{lemma}\n\\begin{proof} Consider a set $D \\subseteq B_{n,m}$ which dominates\n  $I(B_{n,m})$ and achieving $\\ell (D)=b_{n,m}$.  Let $D_P= D\\cap P_{n-12}$,\n  $D_Q= D\\cap Q_{m-12}$, $D_R= D\\cap R_{n-12}$ and $D_O= D\\cap\n  O_{m-12}$. Let $w_P$ ($w_Q$, $w_R$ and $w_O$, respectively) be the\n  input word of $P_{n-12}$ ($Q_{m-12}$, $R_{n-12}$ and $O_{m-12}$), and\n  $w'_P$ ($w'_Q$, $w'_R$ and $w'_O$) be the output word of $P_{n-12}$\n  ($Q_{m-12}$, $R_{n-12}$ and $O_{m-12}$).  By definition of $C_p$,\n  the loss of $D_P$ is at least $C_{n-12}[w_P,w'_P]$. Similarly, we\n  have $\\ell(D_Q)\\geq C_{m-12}[w_Q,w'_Q]$, $\\ell(D_R)\\geq\n  C_{n-12}[w_R,w'_R]$ and $\\ell(D_O)\\geq C_{m-12}[w_O,w'_O]$.\n  By the definition of the loss:\n  \\begin{align} \\ell(D) = {} & b_{n,m} \\nonumber\\\\ = {} & 5\\times\n    \\vert D \\vert - \\vert N[D]\\vert \\nonumber\\\\ = {} & \\ell(D_O)\n    +\\ell(D_P) +\\ell(D_Q) +\\ell(D_R) + L[w'_O,w_P] + L[w'_P,w_Q] +\n    L[w'_Q,w_R] + L[w'_R,w_O] \\nonumber\\\\ & \\text{~~ by\n      Lemma~\\ref{lem:compatible} and since $N[D_P] \\cap N[D_R] = N[D_Q] \\cap\n      N[D_O] =\\emptyset$} \\nonumber\\\\ \\ge {} & C_{m-12}[w_O,w'_O] +\n    C_{n-12}[w_P,w'_P] + C_{m-12}[w_Q,w'_Q] +\n    C_{n-12}[w_R,w'_R] \\nonumber\\\\ & + L[w'_O,w_P] +\n    L[w'_P,w_Q] + L[w'_Q,w_R] + L[w'_R,w_O] \\nonumber\\\\ \\ge {}\n    & M_{m-12}[w_O,w_P] + M_{n-12}[w_P,w_Q] +\n    M_{m-12}[w_Q,w_R] + M_{n-12}[w_R,w_O] \\nonumber \\\\\n    & \\text{~~ since $w'_O$ and $w_P$ (resp. $w'_P$ and $w_Q$, $w'_Q$\n      and $w_R$, $w'_R$ and $w_O$) are compatibles.} \\nonumber\n  \\end{align}\n\\end{proof}\n\n\\medskip\n\nAccording to Lemma~\\ref{lem:bnm}, to bound $b_{n,m}$ it would be thus\ninteresting to know $M_p$ for $p> 12$. It is why we introduce the\nfollowing $8119\\times 8119$ square matrix, $T$.\n\n\\begin{lemma}\n  There exists a matrix $T$ such that $C_{p+1}=C_p \\otimes T$\n  for all $p\\ge 12$. This matrix is defined as follows:\n  $$ T[w,w']=\n  \\begin{cases} \n    +\\infty \\ \\ \\ \\text{ if $\\exists i\\in [10]$ s.t. $w[i]=0$ and $w'[i]=2$}\\\\ \n    +\\infty \\ \\ \\ \\text{ if $\\exists i\\in [9]$ s.t. $w[i]=2$ and $w'[i]\\ne 0$}\\\\ \n    +\\infty \\ \\ \\ \\text{ if $\\exists i\\in \\{2,\\ldots, 9\\}$\n      s.t. $w'[i]=1$, $w[i]\\ne 0$, $w'[i-1]\\ne 0$ and $w'[i+1]\\ne\n      0$}\\\\  \n    +\\infty \\ \\ \\ \\text{ if $w'[1]=1$, $w[1]\\ne 0$ and $w'[2]\\ne 0$}\\\\ \n    +\\infty \\ \\ \\ \\text{ if $w'[10]=1$, $w[10]\\ne 0$ and $w'[9]\\ne 0$}\\\\ \n    3 \\times \\vert w'\\vert_0 - \\vert w\\vert_2 - \\vert w' \\vert_1 +\n    \\vert w\\vert_0 - 1 \\ \\ \\\n    \\text{if $w'[10] = 0$} \\\\\n    3 \\times \\vert w'\\vert_0 - \\vert w\\vert_2 - \\vert w' \\vert_1 +\n    \\vert w\\vert_0 \\ \\ \\ \\text{ otherwise.}\n  \\end{cases}\n  $$\n\\end{lemma}\n\n\\begin{proof} Consider a set $S'\\subseteq P_{p+1}$ dominating\n  $I(P_{p+1})$ and let $S =S' \\cap P_p$.  Let $w = w^{out}_p(S)$ and $w' =\n  w^{out}_{p+1}(S')$.  Let $\\Delta(S,S')= \\los{S'} - \\los{S}$. By\n  definition of the loss, $\\Delta(S,S')= 5 \\times |S'\\setminus S| -\n  |N[S']\\setminus N[S]|$. Let us compute $\\Delta(S,S')$ in term of the\n  number of occurrences of $0$'s, $1$'s and $2$'s in the words $w$ and\n  $w'$. The set $S'\\setminus S$ corresponds to the vertices $\\{(p+1,i)\n  \\mid i\\in[10], w'[i] = 0\\}$. The set $N[S']\\setminus N[S]$ corresponds\n  to the vertices dominated by $S'$ but not dominated by $S$; these\n  vertices clearly belong to the columns $p$, $p+1$ and $p+2$. Since\n  $S'$ dominates $I(P_{p+1})$, those in the column $p$ are the vertices\n  $\\{(p,i) \\mid i\\in[10], w[i]=2\\}$.  Those in the column $p+1$ are the\n  vertices $\\{(p+1,i) \\mid i\\in[10], w'[i]\\neq 2, w[i] \\neq0\\}$ and\n  possibly the vertex $(p+1,11)$ when $w'[10]=0$. Finally, those in the\n  column $p+2$ are the vertices $\\{(p+2,i) \\mid i\\in[10], w'[i]=0\\}$.\n  We then get:\n  $$ \n  \\Delta(S,S')= \n  \\begin{cases}\n    3 \\times \\vert w'\\vert_0 - \\vert w \\vert_2  \n    - \\vert w'\\vert_1 + \\vert w\\vert_0 - 1 & \\text{if $w'[10] = 0$} \\\\    \n    3 \\times \\vert w'\\vert_0 - \\vert w \\vert_2  \n    - \\vert w'\\vert_1 + \\vert w\\vert_0 & \\text{otherwise}\n  \\end{cases}\n  $$\n  where $|w|_n$ denotes the number of occurrences of the letter $n$ in\n  the word $w$.\n\n  Thus $\\Delta(S,S')$ only depends on the output words of $S$ and $S'$,\n  and we can denote this value by $\\Delta(w,w')$. Note however that\n  there exist pairs of\n  words $(w,w')$ that could not be the output words of $S$ and $S'$;\n  there are three cases:\n  \\begin{enumerate}[C{a}se 1.]\n  \\item $w[i]=0$ and $w'[i]=2$ since the vertex\n    $(p+1,i)$ would be dominated by $(p,i)$ contradicting its label 2;\n  \\item $w[i]=2$ and $w'[i]\\ne 0$ for $i\\in[9]$ since \n    $(p,i)$ would not be dominated contradict the fact that $S'$\n    dominates $I(P_{p+1})$;\n  \\item $w'[i]=1$ and $w'[i-1] \\neq 0$, $w'[i+1]\\neq 0$, $w[i]\\neq 0$\n    since $(p+1,i)$ would be dominated according to its label but none\n    of its neighbors belong to $S'$. \n  \\end{enumerate}\n  For these forbidden cases, we set $\\Delta(w,w') = +\\infty$.\n  \n  By definition, $C_{p+1}[w_i,w']$ is the minimum loss $\\ell(S')$\n  of a set $S'\\subseteq P_{p+1}$ that dominates $I(P_{p+1})$, with $w_i$\n  as input word and $w'$ as output word. It is clear that $S= S'\\cap P_p$\n  dominates $I(P_{p})$ and has $w_i$ as input word.  Let $w$ be its output\n  word and note that $C_{p+1}[w_i,w'] = \\ell(S') = \\ell(S) +\n  \\Delta(w_i,w')$.  The minimality of $\\ell(S')$ implies the minimality of\n  $\\ell(S)$ over the sets $X\\in \\mathcal{D}^{w_i,w'}_p$. Indeed, any set $X\\in\n  \\mathcal{D}^{w_i,w'}_p$ could be turned in a set of $X'\\in \\mathcal{D}^{w_i,w'}_{p+1}$ by\n  adding vertices of the $p+1^{th}$ column accordingly to $w'$.  Thus\n  $$\n  C_{p+1}[w_i,w'] = C_{p}[w_i,w] + \\Delta(w,w')\n  $$\n\n  which implies that\n  $$\n  C_{p+1}[w_i,w'] \\geq \\min_{w} C_{p}[w_i,w] + \\Delta(w,w').\n  $$\n\n  On the other hand, for every word $w_o \\in \\mathcal{W}$ such that\n  $C_{p}[w_i,w_o]\\neq +\\infty$ and $\\Delta(w_o,w')\\neq\n  +\\infty$, there is a set $S \\in \\mathcal{D}^{w_i,w_o}_p$, with $\\ell(S) =\n  C_{p}[w_i,w_o]$, that can be turned in a set $S' \\in\n  D^{w_i,w'}_{p+1}$ with $\\ell(S') = C_{p}[w_i,w_o] +\n  \\Delta(w_o,w')$. Thus\n  $$\n  C_{p+1}[w_i,w'] \\leq \\min_{w_o} C_{p}[w_i,w_o] + \\Delta(w_o,w').\n  $$\n\n  This concludes the proof of the lemma.\n\\end{proof}\n\n\nBy the definition of $M_p$, we have also $M_{p+1}=M_p\n\\otimes T$. Note that $T$ is a sparse matrix: about $95.5 \\%$\nof its $8119^2$ entries are $+\\infty$.  Thus the multiplication by\n$T$ in the $(\\min,+)$ algebra can be done in a reasonable amount\nof time by a trivial algorithm.\n\n\\medskip\n\n\\begin{fact}\\label{fact126} The computations give us that\n  $M_{126}=M_{125}+1$. Thus, since $(A + c)\\otimes B =\n  (A\\otimes B) + c$ for any matrices $A$, $B$ and any integer $c$, we\n  have that $M_{125+k}=M_{125}+k$ for every $k\\in \\mathbb{N}$.\n\\end{fact}\n\nLet us define $M'_p= \\min_{k\\in \\mathbb{N}} ( M_{p+k} - k\n)$.  Then, for all $q\\ge p$, $M_q \\ge M'_p + (q-p)$.  By\nFact~\\ref{fact126}, $M'_p= \\min_{k\\in \\{0,\\dots 125-p\\}} (\nM_{p+k} - k )$\n\n\\begin{fact}\\label{fact23} By computing $M'_{12}$, and $A'=\n  M'_{12} \\otimes M'_{12}$, we obtain that $\\min_{w_1,w_3} (A'\n  + A'^T)[w_1,w_3] = 76$ (where $A^T$ is the transpose of $A$).\n\\end{fact}\nThis implies that \n$$ \n\\min_{w_1,w_3} \\ (\\min_{w_2} M'_{12}[w_1,w_2]+M'_{12}[w_2,w_3]) \\ +\n\\ (\\min_{w_4} M'_{12}[w_3,w_4]+ M'_{12}[w_4,w_1]) \\ = \\ 76\n$$\n$$\n\\min_{w_1,w_2,w_3,w_4} M'_{12}[w_1,w_2]+M'_{12}[w_2,w_3]+M'_{12}[w_3,w_4]+\nM'_{12}[w_4,w_1] \\ = \\ 76.\n$$\n\n\\begin{theorem}\n  If $24 \\le n \\le m$, then $$\\gamma(G_{n,m}) = \\form{n}{m}.$$\n\\end{theorem}\n\n\\begin{proof} By Chang's construction \\cite{CHHW}, $\\gamma(G_{n,m}) \\le\n  \\form{n}{m}$. Let us now compute a lower bound for the loss of a\n  dominating set of $G_{n,m}$.\n\n  \\begin{eqnarray*}\n    \\ell_{n,m} &\\ge& b_{n,m}\\\\\n    &\\ge& \\min_{w_1,w_2,w_3,w_4}  M_{n-12}[w_1,w_2] +\n    M_{m-12}[w_2,w_3] + M_{n-12}[w_3,w_4]+\n    M_{m-12}[w_4,w_1]\\\\\n    & & \\; \\mbox{by Lemma~\\ref{lem:bnm}} \\\\\n    &\\ge& \\min_{w_1,w_2,w_3,w_4}  M'_{12}[w_1,w_2]+(n-12-12)  +\n    M'_{12}[w_2,w_3]+ (m-12-12) + M'_{12}[w_3,w_4]\\\\&&\n    \\phantom{\\min_{w_1,w_2,w_3,w_4}}+(n-12-12) +\n    M'_{12}[w_4,w_1]+ (m-12-12)\\\\ \n    &\\ge& 2 \\times (n+m -48) + \\min_{w_1,w_2,w_3,w_4}\n    M'_{12}[w_1,w_2] + M'_{12}[w_2,w_3] +\n    M'_{12}[w_3,w_4] + M'_{12}[w_4,w_1]\\\\ \n    &\\ge& 2 \\times (n+m -48) + 76\\\\ \n    &\\ge& 2\\times(n+m)-20.\n  \\end{eqnarray*}\n\n  Thus by Lemma~\\ref{lemloss}, we have:\n  \\begin{eqnarray*}\n    \\gamma(G_{n,m}) &\\ge& \\left \\lceil \\frac{n\\times m + 2\\times(n+m)-20}{5}\n    \\right \\rceil\\\\\n    &\\ge& \\left \\lceil \\frac{(n+2)(m + 2) -4}{5}\n    \\right \\rceil - 4\\\\\n    &\\ge& \\form{n}{m}.\n  \\end{eqnarray*}\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n                                   {-3.5ex \\@plus -1ex \\@minus -.2ex}%\n                                   {2.3ex \\@plus.2ex}%\n                                   {\\normalfont\\large\\bf}}\n                                 \n\\makeatother\n\n\\makeatletter\n\\renewcommand\\subsection{\\@startsection {subsection}{1}{\\z@}%\n                                   {-3.5ex \\@plus -1ex \\@minus -.2ex}%\n                                   {2.3ex \\@plus.2ex}%\n                                   {\\normalfont\\normalsize\\bf}}\n                                 \n\\makeatother\n\n\n\\usepackage[%\nbookmarks=true,%\nbookmarksdepth=3,%\nbookmarksnumbered=true,%\nsetpagesize=false,%\npdftitle={Local time penalizations\nwith various clocks for L\\'{e}vy processes},%\npdfauthor={Shosei Takeda and Kouji Yano},%\npdfkeywords={one-dimensional L\\'{e}vy process; %\nlimit theorem; penalization; conditioning}%\n]{hyperref}\n\n\\title{\\textbf{Local time penalizations\nwith various clocks for L\\'{e}vy processes}\\footnote{\nThis research was supported by RIMS and by ISM.}}\n\n\\author{Shosei Takeda\\footnote{Rakunan High School, Kyoto, Japan.}\n\\footnote{The research of this author was supported by\nJSPS Open Partnership Joint Research Projects grant no. JPJSBP120209921.}\n\\quad and \\quad Kouji Yano\\footnote{Graduate School of Science, Kyoto University, Japan.}\n\\footnotemark[3]\n\\footnote{The research of this author was supported by\nJSPS KAKENHI grant no.'s JP19H01791 and JP19K21834.}}\n\\date{}\n\\begin{document}\n\\maketitle\n\\begin{abstract}\n  Several long-time limit theorems\n  of one-dimensional L\\'{e}vy processes\n  weighted and normalized\n  by functions of the local time\n  are studied.\n  The long-time limits are taken via certain families of random times, called clocks:\n  exponential clock, hitting time clock, two-point hitting time clock\n  and inverse local time clock.\n  The limit measure can be characterized via a certain martingale\n  expressed by an invariant function for the process\n  killed upon hitting zero.\n  The limit processes may differ according to the choice of the clocks\n  when the original L\\'{e}vy process is recurrent and of finite variance.\n\\end{abstract}\n{\\small Keywords and phrases: one-dimensional L\\'{e}vy process;\nlimit theorem; penalization; conditioning \\\\\nMSC 2020 subject classifications: 60F05 (60G44; 60G51)}\n\\section{Introduction}\nRoynette--Vallois--Yor\n(\\cite{MR2229621,MR2253307} see also~\\cite{MR2261065,MR2504013})\nhave studied the limit distribution for a Brownian motion,\nwhich they called a \\textit{penalization problem},\nas follows.\nLet \\(B=(B_t,t\\ge 0)\\) be a standard Brownian motion and\n\\(L=(L_t,t\\ge 0)\\) denote its local time at \\(0\\). Then, for\nany positive integrable function \\(f\\)\nand any bounded adapted functional \\(F_t\\), it holds that\n\\begin{align}\n  \\lim_{s\\to\\infty} \\frac{\\ensuremath{\\mathbb{P}}\\sbra{F_t f(L_s)}}{\\ensuremath{\\mathbb{P}}\\sbra{f(L_s)}}\n  = \\ensuremath{\\mathbb{P}}\\sbra*{F_t \\frac{M_t}{M_0}} \\eqqcolon \\ensuremath{\\mathbb{Q}}\\sbra{F_t},\n\\end{align}\nwhere \\(M=(M_t,t\\ge 0)\\) is the martingale given by\n\\begin{align}\n  M_t = f(L_t)\\abs{B_t} + \\int_0^\\infty f(L_t + u)\\, \\mathrm{d} u,\n  \\quad t\\ge 0.\n\\end{align}\nUnder the penalized probability\nmeasure \\(\\ensuremath{\\mathbb{Q}}\\), the total local time \\(L_{\\infty}\\) is finite, and\nin fact, a sample path behaves as the concatenation of a Brownian bridge\nand a three-dimensional Bessel process; see~\\cite{MR2229621}.\nIn particular, \\(\\ensuremath{\\mathbb{Q}}\\) is singular to \\(\\ensuremath{\\mathbb{P}}\\).\n\nThis result for a Brownian motion was\ngeneralized to many other processes.\nIn particular, we refer to\nDebs~\\cite{MR2599215} for random walks,\nNajnudel--Roynette--Yor~\\cite{MR2528440} for Markov chains and Bessel processes,\nYano--Yano--Yor~\\cite{MR2552915} for symmetric stable processes,\nSalminen--Vallois~\\cite{MR2540855} and Profeta~\\cite{MR2760742,MR2968676} for linear\ndiffusions.\nMost of these results were obtained basically under the assumption of some\nregular variation condition.\nProfeta--Yano--Yano~\\cite{MR3909919} developed a general theory for\none-dimensional diffusions by adopting a random clock approach.\nThey studied the long-time limit of the form\n\\begin{align}\n  \\lim_{\\tau\\to\\infty}\\frac{\\ensuremath{\\mathbb{P}}\\sbra{F_t f(L_\\tau)}}{\\ensuremath{\\mathbb{P}}\\sbra{f(L_\\tau)}},\n\\end{align}\nwhere \\(\\tau=(\\tau_\\lambda)\\) is a certain parametrized\nfamily of random times, which they called a clock.\nSuch a random clock approach already\nappeared in the problem of conditioning to avoid zero,\nwhich is a special case of our penalization with\n\\(f(u)=1_{\\cbra{u=0}}\\), or in the problem of conditioning\nto stay positive\/negative.\nFor example, we refer to Knight~\\cite{MR253424} for Brownian motions,\nChaumont~\\cite{MR1419491}, Chaumont--Doney~\\cite{MR2164035,MR2375597}\nand Doney~\\cite[Section 8]{MR2126962,MR3155252}\nfor L\\'{e}vy processes conditioned to stay positive,\nYano--Yano~\\cite{MR3444297} for diffusions and\nPant\\'{\\i}~\\cite{MR3689384} for L\\'{e}vy processes conditioned to avoid zero.\n\nLet \\(X=(X_t,t\\ge 0)\\) be a one-dimensional L\\'{e}vy process and\nlet \\(T_A\\) denote the hitting time of a Borel\nset \\(A \\subset \\ensuremath{\\mathbb{R}}\\) for \\(X\\), i.e.,\n\\begin{align}\\label{eq:T_A}\n  T_A = \\inf\\cbra{t>0\\colon X_t \\in A}\n\\end{align}\nand we write \\(T_a = T_{\\cbra{a}}\\)\nsimply for the hitting time of a point \\(a\\in\\ensuremath{\\mathbb{R}}\\).\nLet \\((\\eta^a_u)\\) denote the right-continuous inverse of\nthe local time at a point \\(a\\in \\ensuremath{\\mathbb{R}}\\).\nWe adopt the random clock approach for the following four clocks:\n\\begin{enumerate}\n  \\item exponential clock: \\(\\tau=(\\bm{e}_q)\\) with \\(q\\to 0+\\);\n  \\item hitting time clock: \\(\\tau=(T_a)\\) with\n        \\(a\\to\\pm\\infty\\);\\label{item:hitting-time-clocks}\n  \\item two-point hitting time clock: \\(\\tau=(T_{a}\\wedge T_{-b})\\) with\n        \\(a\\to\\infty\\) and \\(b\\to\\infty\\);\\label{item:two-point-hitting}\n  \\item inverse local time clock: \\(\\tau=(\\eta^a_u)\\)\n        with \\(a\\to\\pm\\infty\\) or with \\(u\\to\\infty\\).\n\\end{enumerate}\n\n\\subsection{Main results}\\label{Subsec:main-results}\nLet \\((X, \\ensuremath{\\mathbb{P}}_x)\\) denote the canonical representation\nof a L\\'{e}vy process starting from \\(x\\)\non the c\\`{a}dl\\`{a}g path space \\(\\ensuremath{\\mathcal{D}}\\)\nand set \\(\\ensuremath{\\mathbb{P}}=\\ensuremath{\\mathbb{P}}_0\\).\nFor \\(t\\ge 0\\), we denote by \\(\\ensuremath{\\mathcal{F}}_t^X = \\sigma(X_s, 0\\le s\\le t)\\) the natural\nfiltration of \\(X\\) and write \\(\\ensuremath{\\mathcal{F}}_t = \\bigcap_{s>t} \\ensuremath{\\mathcal{F}}_s^X\\).\nWe have\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{\\mathrm{i}\\lambda X_t}} = \\mathrm{e}^{-t \\varPsi(\\lambda)},\n  \\quad t \\ge 0, \\,\\lambda \\in \\ensuremath{\\mathbb{R}},\n\\end{align}\nwhere \\(\\varPsi(\\lambda)\\) denotes the characteristic exponent of \\(X\\)\ngiven by the L\\'{e}vy--Khintchine formula\n\\begin{align}\n  \\varPsi(\\lambda)\n  = \\mathrm{i} v \\lambda\n  + \\frac{1}{2} \\sigma^2 \\lambda^2\n  + \\int_\\ensuremath{\\mathbb{R}} \\rbra*{1 - \\mathrm{e}^{\\mathrm{i} \\lambda x} + \\mathrm{i} \\lambda x 1_{\\cbra{\\abs{x}< 1}}}\n  \\nu(\\mathrm{d} x)\n\\end{align}\nfor some constants \\(v \\in \\ensuremath{\\mathbb{R}}\\) and \\(\\sigma \\ge 0\\)\nand some measure \\(\\nu\\) on \\(\\ensuremath{\\mathbb{R}}\\)\n(called the L\\'{e}vy measure)\nwhich satisfies \\(\\nu(\\cbra{0})=0\\) and\n\\begin{align}\n  \\int_\\ensuremath{\\mathbb{R}} \\rbra*{x^2 \\wedge 1} \\nu(\\mathrm{d} x) < \\infty.\n\\end{align}\nWe denote the real and imaginary parts of \\(\\varPsi(\\lambda)\\) by\n\\begin{align}\n  \\theta(\\lambda) & = \\Re \\varPsi(\\lambda)\n  = \\frac{1}{2} \\sigma^2 \\lambda^2\n  + \\int_\\ensuremath{\\mathbb{R}} \\rbra*{1 - \\cos \\lambda x} \\nu(\\mathrm{d} x),\\label{eq:theta} \\\\\n  \\omega(\\lambda) & = \\Im \\varPsi(\\lambda)\n  = v \\lambda\n  + \\int_\\ensuremath{\\mathbb{R}} \\rbra*{\\lambda x 1_{\\cbra{\\abs{x}< 1}} - \\sin \\lambda x} \\nu(\\mathrm{d}\n  x)\\label{eq:omega}.\n\\end{align}\nNote that \\(\\theta(\\lambda) \\ge 0\\) for \\(\\lambda \\in \\ensuremath{\\mathbb{R}}\\),\n\\(\\theta(\\lambda)\\) is even and \\(\\omega(\\lambda)\\) is odd.\nFor more details of the notation of this section,\nsee Section~\\ref{Sec:preliminaries}.\nThroughout this paper except\nSections~\\ref{Sec:preliminaries},~\\ref{Sec:trans} and~\\ref{Sec:mart},\nwe always assume\n\\((X,\\ensuremath{\\mathbb{P}})\\) is recurrent, i.e.,\n\\begin{align}\\label{eq:recurrent}\n  \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty 1_{\\cbra{\\abs{X_t-a}<\\varepsilon}}\\, \\mathrm{d} t}\n  =\\infty, \\qquad\\text{for all \\(a\\in\\ensuremath{\\mathbb{R}}\\) and \\(\\varepsilon>0\\),}\n\\end{align}\n and assume the following:\n\\begin{enumerate}[label=\\textbf{(\\Alph*)}]\n  \\item For each \\(q > 0\\), it holds that\n        \\begin{align}\n          \\int_0^\\infty\n          \\abs*{\\frac{1}{q+\\varPsi(\\lambda)}}\n          \\, \\mathrm{d} \\lambda < \\infty.\n        \\end{align}\\label{item:assumption}\n\\end{enumerate}\nNote that, we say \\((X,\\ensuremath{\\mathbb{P}})\\) is transient if~\\eqref{eq:recurrent}\ndoes not hold.\nUnder the assumption~\\ref{item:assumption},\nthe process\n\\((X,\\ensuremath{\\mathbb{P}})\\) is recurrent if and only if \\((X,\\ensuremath{\\mathbb{P}})\\) is\npoint recurrent, i.e.,\n\\begin{align}\\label{eq:pt-recurrent}\n  \\ensuremath{\\mathbb{P}}(T_a<\\infty) = 1,\\qquad\\text{for all \\(a\\in\\ensuremath{\\mathbb{R}}\\);}\n\\end{align}\nsee~Subsection~\\ref{Subsec:LT-em}.\nThe assumption~\\ref{item:assumption}\nimplies that the \\(q\\)-resolvent density \\(r_q\\) exists for \\(q>0\\);\nsee Subsection~\\ref{Subsec:resolvent}.\nFor \\( q > 0\\), we define\n\\begin{align}\\label{eq:def-h_q}\n  h_q(x) = r_q(0) - r_q(-x)\n  = \\frac{1}{\\pi}\n  \\int_0^\\infty\n  \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{q+\\varPsi(\\lambda)}}\n  \\,\\mathrm{d} \\lambda\n  , \\qquad x \\in \\ensuremath{\\mathbb{R}},\n\\end{align}\nwhere the second identity follows from Proposition~\\ref{Prop:r_q-represent}.\nIt is obvious that \\(h_q(0) = 0\\), and\nby~\\eqref{eq:exp-hit-time}, we have \\(h_q(x) \\ge 0\\).\nWe denote the second moment by\n\\begin{align}\n  m^2 =\n  \\ensuremath{\\mathbb{P}}\\sbra{{X_1}^2} \\in (0, \\infty].\\label{eq:second-moment}\n\\end{align}\nThe following theorem plays a key role\nin our penalization results.\nRecall that \\(X\\) is assumed recurrent.\n\n\\begin{Thm}\\label{Thm:exist-h}\n  Suppose that~\\ref{item:assumption}\n  is satisfied.\n  Then the following assertions hold.\n  \\begin{enumerate}\n    \\item For any \\(x \\in \\ensuremath{\\mathbb{R}}\\),\n          \\begin{align}\\label{eq:conv-h}\n            h(x) \\coloneqq \\lim_{q \\to 0+} h_q(x)\n          \\end{align}\n          exists and is finite,\n          which will be called the \\emph{renormalized zero resolvent}.\n          If \\(m^2 < \\infty\\), then \\(h\\) has the following representation:\n          \\begin{align}\\label{eq:h-repre}\n            h(x)\n            = \\frac{1}{\\pi} \\int_0^\\infty \\Re\\rbra*{\n              \\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{\\varPsi(\\lambda)}}\n            \\, \\mathrm{d} \\lambda.\n          \\end{align}\\label{Thm-item:exist-h}\n    \\item The convergence~\\eqref{eq:conv-h} is uniform on compacts, and\n          consequently \\(h\\) is continuous.\\label{Thm-item:unif-conv-h}\n    \\item \\(h\\) is subadditive on \\ensuremath{\\mathbb{R}},\n          that is, \\(h(x+y)\\le h(x)+h(y) \\) for \\(x,y\\in\n          \\ensuremath{\\mathbb{R}}\\).\\label{Thm-item:subadditive-h}\n  \\end{enumerate}\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:exist-h} will be given in\nSection~\\ref{Subsec:pf-h}.\nThe renormalized zero resolvent satisfies the following limit properties.\n\\begin{Thm}\\label{Thm:property-h}\n  Suppose that~\\ref{item:assumption}\n  is satisfied.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item\n          \\(\\displaystyle\n          \\lim_{x\\to\\pm\\infty} \\frac{h(x)}{\\abs{x}} =\n          \\frac{1}{m^2} \\in [0,\\infty)\n          \\);\\label{Thm-item:h\/x-infinity}\n    \\item\n          \\(\\displaystyle\\label{eq:h-h-limit}\n          \\lim_{y\\to\\pm\\infty} \\cbra*{h(x+y) - h(y)} =\n          \\pm\\frac{x}{m^2}\\in\\ensuremath{\\mathbb{R}},\n          \\) for all \\(x\\in\\ensuremath{\\mathbb{R}}\\).\\label{Thm-item:h-h-limit}\n  \\end{enumerate}\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:property-h}\nwill be given in Section~\\ref{Subsec:pf-h}.\n\n\\begin{Cor}\n  Suppose that~\\ref{item:assumption}\n  is  satisfied.\n  For \\(-1\\le \\gamma\\le 1\\), define\n  \\begin{align}\\label{eq:def-h-gamma}\n    h^{(\\gamma)}(x)=h(x)+\\frac{\\gamma}{m^2}x, \\quad x\\in\\ensuremath{\\mathbb{R}}.\n  \\end{align}\n  Then \\(h^{(\\gamma)}\\) is subadditive and \\(h^{(\\gamma)}(x)\\ge 0\\).\n\\end{Cor}\n\\begin{proof}\n  By definition, we have \\(h^{(\\gamma)}(0)=0\\).\n  From (\\ref{Thm-item:subadditive-h}) of Theorem~\\ref{Thm:exist-h},\n  the function \\(h^{(\\gamma)}\\) is subadditive. From\n  (\\ref{Thm-item:h\/x-infinity}) of Theorem~\\ref{Thm:property-h},\n  it holds that \\(\\lim_{x\\to\\pm\\infty} h^{(\\gamma)}(x)\/\\abs{x} = (1+\\gamma)\/m^2 \\ge\n  0\\).\n  Suppose \\(h^{(\\gamma)}(x)<0\\) for some \\(x\\in\\ensuremath{\\mathbb{R}}\\setminus\\cbra{0}\\).\n  Since \\(h^{(\\gamma)}\\) is subadditive,\n  we have \\(h(nx)\/n\\le h(x)<0\\) for \\(n=1,2,\\dots\\).\n  Letting \\(n\\to\\infty\\), we face the contradiction.\n  Therefore we have \\(h^{(\\gamma)}(x)\\ge 0\\) for all \\(x\\in\\ensuremath{\\mathbb{R}}\\).\n\\end{proof}\nWe will prove in Theorem~\\ref{Thm:hg-invariant} that\nthe function \\(h^{(\\gamma)}\\) is invariant for the process\nkilled upon hitting zero.\nLet \\(\\ensuremath{\\mathcal{L}}_{+}^1\\) denote the set of non-negative functions\non \\([0,\\infty)\\) which satisfy \\(\\int_0^\\infty f(x)\\,\\mathrm{d} x<\\infty\\).\nFor \\(f\\in \\ensuremath{\\mathcal{L}}_{+}^1\\), define\n\\begin{align}\\label{eq:def-mart}\n  M_t^{(\\gamma)}= M_t^{(\\gamma, f)}= h^{(\\gamma)}(X_t)f(L_t)+\n  \\int_0^\\infty\n  f(L_t+u)\\,\\mathrm{d} u.\n\\end{align}\nNote that, when \\(m^2=\\infty\\),\nwe have \\(h^{(\\gamma)}=h^{(0)}=h\\) and\n\\(M^{(\\gamma)}=M^{(0)}\\) for all \\(\\gamma\\).\n\\begin{Thm}\\label{Thm:martingale}\n  Suppose that~\\ref{item:assumption}\n  is satisfied.\n  Let \\(f\\in \\ensuremath{\\mathcal{L}}^1_{+}\\), \\(-1\\le \\gamma\\le 1\\) and \\(x\\in \\ensuremath{\\mathbb{R}}\\). Then\n  \\((M_t^{(\\gamma)},t\\ge 0)\\) is a non-negative \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{Thm}\nTheorem~\\ref{Thm:martingale} will be proved in\nSection~\\ref{Subsec:pf-hitting-l1}.\nUsing this martingale,\nwe discuss our penalization problems.\nLet \\(L=(L_t)\\) denote the local time at the origin of \\(X\\);\nsee Section~\\ref{Subsec:LT-em}.\n\n\\begin{Thm}[hitting time clock]\\label{Thm:hitting-time-result}\n  Suppose that the condition~\\ref{item:assumption} is satisfied.\n  Let \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\) and \\(x \\in \\ensuremath{\\mathbb{R}}\\).\n  Define \\(h^B(a)=\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}\\) and\n  \\begin{align}\n    N_t^a & = h^B(a) \\ensuremath{\\mathbb{P}}_x\\sbra*{f(L_{T_a}); t<T_a|\\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^a & = h^B(a) \\ensuremath{\\mathbb{P}}_x\\sbra*{f(L_{T_a})|\\ensuremath{\\mathcal{F}}_t}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\lim_{a\\to\\pm\\infty}N_t^a\n    =\\lim_{a\\to\\pm\\infty} M_t^a\n    = M_t^{(\\pm 1)},\n    \\qquad \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n  Consequently, if \\(M_0^{(\\pm 1)}>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{T_a})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t^{(\\pm 1)}}{M_0^{(\\pm 1)}}},\n    \\qquad \\text{as \\(a \\to \\pm\\infty \\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\nTheorem~\\ref{Thm:hitting-time-result} will be proved\nin Section~\\ref{Sec:hitting-time}.\nIf we take \\(f=1_{\\cbra{u=0}}\\), we obtain\nthe conditioning result.\n\\begin{Cor}\\label{Cor:hitting-cond}\nSuppose that the condition~\\ref{item:assumption} is satisfied.\nLet \\(x\\in \\ensuremath{\\mathbb{R}}\\) with \\(h^{(\\pm 1)}(x)>0\\).\nThen it holds that\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>T_a}\n  \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t\\frac{h^{(\\pm 1)}(X_t)}{h^{(\\pm 1)}(x)};T_0>t},\n  \\qquad\\text{as \\(a\\to\\pm\\infty\\),}\n\\end{align}\nfor all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Cor}\nSee also Corollary~\\ref{Cor:avoid-zero}.\n\nLet us state our penalization result with two-point hitting time clock.\nFor \\(a,b\\in\\ensuremath{\\mathbb{R}}\\), we write\n\\(T_{a,b} = T_{\\cbra{a,b}}= T_{a} \\wedge T_{b}\\).\nFor \\(-1\\le\\gamma\\le 1\\), we say\n\\begin{align}\\label{eq:a-b-inf-notation}\n  \\text{\\((a,b)\\xrightarrow[]{\\gamma}\\infty\\)\n    when \\(a\\to\\infty\\), \\(b\\to\\infty\\) and \\(\\frac{a-b}{a+b}\\to\\gamma\\).}\n\\end{align}\n\\begin{Thm}[two-point hitting time clock]\\label{Thm:two-point-hitting-time-result}\n  Suppose that the condition~\\ref{item:assumption} is satisfied.\n  Let \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\), \\(x \\in \\ensuremath{\\mathbb{R}}\\), and \\(a,b>0\\).\n  Define \\(\th^C(a,-b) = \\ensuremath{\\mathbb{P}}\\sbra{L_{T_{a, -b}}}\\), and\n  \\begin{align}\n    N_t^{a,b}\n     & =h^C(a, -b)\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_{a, -b}}); t<T_{a,-b}|\\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^{a,b}\n     & =h^C(a, -b)\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_{a,-b}})|\\ensuremath{\\mathcal{F}}_t}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\lim_{(a,b)\\xrightarrow[]{\\gamma}\\infty}N_t^{a,b}\n    =\\lim_{(a,b)\\xrightarrow[]{\\gamma}\\infty} M_t^{a,b}\n    =M_t^{(\\gamma)},\\quad\n    \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and\n      in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n  Consequently, if \\(M_0^{(\\gamma)}>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{T_{a, -b}})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_{a,-b}})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t^{(\\gamma)}}{M_0^{(\\gamma)}}},\n    \\qquad\n    \\text{as \\((a,b)\\xrightarrow[]{\\gamma}\\infty\\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:two-point-hitting-time-result}\nwill be given in Section~\\ref{Subsec:pf-two-hitting}.\n\\begin{Cor}\\label{Cor:two-point-hitting-cond}\nSuppose that the condition~\\ref{item:assumption} is satisfied.\nLet \\(-1\\le\\gamma\\le 1\\) and\n\\(x\\in \\ensuremath{\\mathbb{R}}\\) with \\(h^{(\\gamma)}(x)>0\\).\nThen it holds that\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>T_{a,-b}}\n  \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t\\frac{h^{(\\gamma)}(X_t)}{h^{(\\gamma)}(x)};T_0>t},\n  \\qquad\\text{as \\((a,b)\\xrightarrow[]{\\gamma}\\infty\\),}\n\\end{align}\nfor all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Cor}\nSee also Corollary~\\ref{Cor:avoid-zero}.\n\nNote that Theorems~\\ref{Thm:hitting-time-result}\nand~\\ref{Thm:two-point-hitting-time-result} show that the limit\nlaw varies according to the chosen clock\nwhen \\(m^2<\\infty\\).\n\n\\subsection{Backgrounds of the renormalized zero resolvent}\\label{Subsec:back-h}\nThe existence of \\(h\\) for symmetric L\\'{e}vy processes\nwas proved by Salminen--Yor~\\cite{MR2409011}\nunder the assumption~\\ref{item:assumption}; see also\nYano~\\cite{MR2603019}.\nWe shall review early studies of the existence of \\(h\\) and\nits limit properties\nfor asymmetric processes.\n\nSimilar results were obtained for random walks\nby Spitzer~\\cite[Chapter VII]{MR0171290},\nPort--Stone~\\cite{MR0215375,MR0226740,MR261706}\nand Stone~\\cite{MR238398}.\nFor L\\'{e}vy processes, Port--Stone~\\cite[Section 17]{MR346919} obtained some results\nwhich were similar to but different from\nTheorems~\\ref{Thm:exist-h} and~\\ref{Thm:property-h},\nreducing them to the random walk case.\n(For the proofs\nof Theorems~\\ref{Thm:exist-h} and~\\ref{Thm:property-h},\nwe are inspired\nby Spitzer~\\cite[Chapter VII]{MR0171290},\nPort--Stone~\\cite[Section 17]{MR0215375,MR0226740,MR261706,MR346919}\nand Stone~\\cite{MR238398}.)\n\nYano~\\cite{MR3072331} showed\nthe existence of the renormalized zero resolvent \\(h\\)\nunder the following two conditions:\n\\begin{enumerate}[label=\\textbf{(Y\\arabic*)}]\n  \\item \\(\\displaystyle \\int_0^\\infty \\frac{1}{q+\\theta(\\lambda)}\n        \\, \\mathrm{d} \\lambda < \\infty\\) for all \\(q>0\\);\\label{item:cond-Yano1}\n  \\item \\(\\theta\\) and \\(\\omega\\) have measurable derivatives\n        on \\((0, \\infty)\\) which satisfy\n        \\begin{align}\n          \\int_0^\\infty \\frac{\\rbra{\n            \\abs{\\theta^\\prime(\\lambda)}+\n            \\abs{\\omega^\\prime(\\lambda)}}(\\lambda^2\\wedge 1)\n          }{{\\theta(\\lambda)}^2+{\\omega(\\lambda)}^2}\n          \\, \\mathrm{d} \\lambda < \\infty.\n        \\end{align}\\label{item:cond-Yano2}\n\\end{enumerate}\nPant\\'{\\i}~\\cite{MR3689384} proved the existence of \\(h\\)\nunder the condition\n\\begin{enumerate}[label=\\textbf{(P)}]\n  \\item \\ref{item:cond-Yano1} and \\( \\displaystyle\n        \\int_\\ensuremath{\\mathbb{R}} \\abs*{\\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda}}{\\varPsi(\\lambda)}}}\n        \\, \\mathrm{d} \\lambda < \\infty\\),\\label{item:Panti-cond}\n\\end{enumerate}\nwhich is weaker than~\\ref{item:cond-Yano1} and~\\ref{item:cond-Yano2},\nand applied it to the conditioning to avoid zero\nwith exponential clock.\nTsukada~\\cite{MR3838874} also proved the existence\nof \\(h\\)\nunder the assumption\n\\begin{enumerate}[label=\\textbf{(T)}]\n  \\item \\ref{item:assumption} and\n        \\(\\displaystyle\n        \\int_0^1 \\abs*{\\Im\\rbra*{\\frac{\\lambda}{\\varPsi(\\lambda)}}}\n        \\, \\mathrm{d} \\lambda <\n        \\infty,\n        \\)\\label{item:Tsukada-b}\n\\end{enumerate}\nwhich is weaker than~\\ref{item:Panti-cond};\nsee~\\cite[Proposition 15.3]{MR3838874}.\n\n\\begin{Rem}\n  We do not know whether the integral\n  representation~\\eqref{eq:h-repre} also holds\n  in the case \\(m^2=\\infty\\).\n  Yano~\\cite{MR3072331} showed that, if \\(X\\) is symmetric,\n  then~\\eqref{eq:h-repre} holds.\n  Tsukada~\\cite{MR3838874} showed that~\\ref{item:Tsukada-b} implies\n  \\eqref{eq:h-repre}.\n\\end{Rem}\n\n\\subsection{Organization}\nThe remainder of this paper is organized as follows.\nIn Section~\\ref{Sec:preliminaries}, we prepare\ncertain general properties and\npreliminary facts of L\\'{e}vy processes.\nIn Section~\\ref{Sec:h}, we\nstudy the renormalized zero resolvent.\nIn Sections~\\ref{Sec:exp-clock},~\\ref{Sec:hitting-time},~\\ref{Sec:two-hitting-time}\nand~\\ref{Sec:inv-time}, we discuss the penalization results\nwith exponential clock, hitting time clock, two-point hitting time\nclock and inverse local time clock, respectively.\nIn Section~\\ref{Sec:universal-measure}, we introduce certain universal\n\\(\\sigma\\)-finite measures to study long time\nbehaviors of sample paths of the penalized measure.\nIn Section~\\ref{Sec:trans}, we study penalization in the transient case.\nIn Section~\\ref{Sec:mart} as an appendix, we study martingale property\nof \\((X_t f(L_t),t\\ge 0)\\).\n\n\\subsection*{Acknowledgments}\nThe authors would like to thank Hiroshi Tsukada\nfor his helpful comments.\n\n\\section{Preliminaries}\\label{Sec:preliminaries}\n\\subsection{Absolutely continuous resolvent}\\label{Subsec:resolvent}\nWe now consider the following two conditions:\n\\begin{enumerate}[label=\\textbf{(A\\arabic*)},series=cond]\n  \\item\\label{item:cond-not-CPP}\n        The process \\(X\\) is not a compound Poisson process;\n  \\item\\label{item:cond-regular}\n        \\(0\\) is regular for itself, i.e.,\n        \\(\\ensuremath{\\mathbb{P}}(T_0 = 0) = 1\\).\n\\end{enumerate}\n\nThe next lemma is due to Kesten~\\cite{MR0272059} and\nBretagnolle~\\cite{MR0368175}.\n\\begin{Lem}[\\cite{MR0272059,MR0368175}]\\label{Lem:iff-cond-notCPP-regular}\n  The conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular}\n  hold if and only if the following two assertions hold:\n  \\begin{enumerate}[resume*=cond]\n    \\item For each \\(q>0\\),\n          the characteristic exponent \\(\\varPsi\\) satisfies\n          \\begin{align}\n            \\int_\\ensuremath{\\mathbb{R}} \\Re\\rbra*{\\frac{1}{q+\\varPsi(\\lambda)}}\\, \\mathrm{d} \\lambda < \\infty;\n          \\end{align}\\label{item:cond-Reinte}\n    \\item We have either \\(\\sigma>0\\) or\n          \\(\\int_{(-1, 1)} \\abs{x} \\nu(\\mathrm{d} x) = \\infty\\).\\label{item:cond-nu}\n  \\end{enumerate}\n  Furthermore, under the condition~\\ref{item:cond-Reinte},\n  the condition~\\ref{item:cond-regular} holds if and only\n  if the condition~\\ref{item:cond-nu} holds.\n\\end{Lem}\nIf the above conditions hold, it is known that\n\\(X\\) has a bounded continuous resolvent density. See,\ne.g., Theorem II.16 and Theorem II.19 of Bertoin~\\cite{MR1406564}\n\\begin{Lem}[\\cite{MR1406564}]\\label{Lem:resolvent-density}\n  The condition~\\ref{item:cond-Reinte} holds if and only if\n  \\(X\\) has the bounded \\(q\\)-resolvent density \\(r_q\\), for \\( q > 0\\),\n  which satisfies\n  \\begin{align}\n    \\int_\\ensuremath{\\mathbb{R}} f(x) r_q(x) \\, \\mathrm{d} x\n    = \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt}f(X_t) \\, \\mathrm{d} t}\n  \\end{align}\n  for all non-negative measurable functions \\(f\\).\n  Moreover, under the condition~\\ref{item:cond-Reinte},\n  the condition~\\ref{item:cond-regular} holds if and only if\n  \\(x \\mapsto r_q(x)\\) is continuous.\n\\end{Lem}\n\nIf \\(r_q(x)\\) is bounded in \\(x\\in\\ensuremath{\\mathbb{R}}\\),~\\cite[Corollary II.18]{MR1406564}\nimplies that\nthe Laplace transform of \\(T_0\\)\ncan be represented as\n\\begin{align}\\label{eq:exp-hit-time}\n  \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_0}} = \\frac{r_q(-x)}{r_q(0)}, \\quad q>0, \\; x \\in \\ensuremath{\\mathbb{R}}.\n\\end{align}\n\n\\begin{Prop}\\label{Prop:r_q-represent}\n  Suppose that the condition~\\ref{item:assumption} holds.\n  Then the bounded continuous resolvent density can be expressed as\n  \\begin{align}\n    r_q(x) =\n    \\frac{1}{2\\pi}\\int_{-\\infty}^\\infty\n    \\frac{\\mathrm{e}^{-\\mathrm{i}\\lambda x}}{q+\\varPsi(\\lambda)}\n    \\, \\mathrm{d} \\lambda\n    =\\frac{1}{\\pi}\\int_0^\\infty\n    \\Re\\rbra*{\\frac{\\mathrm{e}^{-\\mathrm{i}\\lambda x}}{q+\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n  \\end{align}\n  for all \\(q > 0\\) and \\(x\\in \\ensuremath{\\mathbb{R}}\\).\n\\end{Prop}\nProposition~\\ref{Prop:r_q-represent} can be proved using Fourier\ninversion formula; see, e.g.,~\\cite[Lemma 2]{MR1894112}\nand~\\cite[Corollary 15.1]{MR3838874}.\nBy Lemmas~\\ref{Lem:iff-cond-notCPP-regular}\nand~\\ref{Lem:resolvent-density} and Proposition~\\ref{Prop:r_q-represent},\nthe condition\n~\\ref{item:assumption}\nimplies~\\ref{item:cond-not-CPP}--\\ref{item:cond-nu}.\n\\begin{Lem}[Tsukada {\\cite[Lemma 15.5]{MR3838874}}]\\label{Lem:inte-of-varPsi}\n  Suppose that the condition~\\ref{item:assumption} holds.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item \\(\\abs{\\varPsi(\\lambda)} \\to \\infty\\)\n          as \\(\\lambda\\to\\pm\\infty\\);\\label{item:varpsi-to-infty}\n    \\item \\(\\displaystyle \\int_\\delta^\\infty \\abs*{\\frac{1}{\\varPsi(\\lambda)}}\n          \\, \\mathrm{d} \\lambda < \\infty\\)\n          for all \\(\\delta>0\\);\\label{Lem-item:infty-inte}\n    \\item \\(\\displaystyle \\int_0^\\delta \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}\n          \\, \\mathrm{d} \\lambda < \\infty\\)\n          for all \\(\\delta>0\\);\\label{Lem-item:zero-inte}\n    \\item \\(\\displaystyle \\lim_{q\\to 0+}\n          \\int_0^\\infty \\abs*{\\frac{q}{q+\\varPsi(\\lambda)}}\n          \\, \\mathrm{d} \\lambda\n          = 0\\).\n          In particular, \\(qr_q(x) \\to 0\\) as \\(q \\to 0+\\).\\label{Lem-item:qr_q}\n  \\end{enumerate}\n\\end{Lem}\n\n\\subsection{Local time and its excursion measure}\\label{Subsec:LT-em}\nAssume the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\nThen we can define local time at \\(0\\), which we denote by \\(L=(L_t,t\\ge 0)\\).\nNote that \\(L\\) is continuous in \\(t\\) and satisfies\n\\begin{align}\\label{eq:regularity-of-L}\n  \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt} \\mathrm{d} L_t} = r_q(-x),\n  \\quad q>0,\\;x \\in \\ensuremath{\\mathbb{R}}.\n\\end{align}\nSee, e.g.,~\\cite[Section V]{MR1406564}.\nIn particular, \\(r_q(x)\\) is non-decreasing as \\(q \\to 0+\\).\nLet \\(\\eta=\\rbra{\\eta_l, l \\ge 0}\\) denote\nthe right-continuous inverse of \\(L\\)\nwhich is given as\n\\(\\eta_l = \\inf\\cbra{t > 0\\colon L_t > l}\\).\nThen the process \\((\\eta, \\ensuremath{\\mathbb{P}})\\) is a possibly killed\nsubordinator, and its Laplace transform is\n\\(\\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{-q\\eta_l}} = \\mathrm{e}^{-l\/r_q(0)}\\), for\n\\(l,q > 0\\),\nsee, e.g.,~\\cite[Proposition V.4]{MR1406564}.\n\nNow we can apply It\\^{o}'s excursion theory.\nLet \\(n\\) denote the characteristic measure of excursions away from the origin.\nWe denote \\(e_l\\) for excursion which starts at local time \\(l\\).\nThen we see that the subordinator \\(\\eta\\) has no drift and its L\\'{e}vy measure is\n\\(n(T_0\\in \\mathrm{d} x)\\).\nIn particular, we have\n\\begin{align}\\label{eq:exp-nt0}\n  \\mathrm{e}^{-l\/r_q(0)} = \\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{-q\\eta_l}}\n  = \\exp\\rbra{-l n\\sbra{1-\\mathrm{e}^{-qT_0}}},\n  \\quad l \\ge 0.\n\\end{align}\nThis implies that\n\\begin{align}\\label{eq:rel-n-r_q}\n  n\\sbra{1 - \\mathrm{e}^{-qT_0}} = \\frac{1}{r_q(0)},\n\\end{align}\nwhich is also obtained from~\\cite[(3.16)]{MR2552915}.\nNow set\n\\begin{align}\n  \\kappa = \\lim_{q\\to 0+}\\frac{1}{r_q(0)} = n(T_0 =\\infty).\\label{eq:kappa}\n\\end{align}\nIt is known that \\(\\kappa = 0\\) (resp.\\ \\(\\kappa > 0\\))\nif and only if \\(X\\) is recurrent (resp.\\ transient);\nsee, e.g.,~\\cite[Theorem I.17]{MR1406564}\nand~\\cite[Theorem 37.5]{MR1739520}.\nIt is also known that\n\\(X\\) is recurrent if and only if \\(X\\) is point recurrent;\nsee, e.g.,~\\cite[Remark 43.12]{MR1739520}\n(see also~\\cite[Excercise II.6.4]{MR1406564}).\nUnder the assumption~\\ref{item:assumption},\nwe can prove this fact by using Theorem~\\ref{Thm:exist-h};\nin fact,\nby equations~\\eqref{eq:exp-hit-time} and~\\eqref{eq:kappa},\nit holds that, for \\(x\\in\\ensuremath{\\mathbb{R}}\\),\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_x(T_0<\\infty) &= \\lim_{q\\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_0}}\n  = \\lim_{q\\to 0+}\\frac{r_q(-x)}{r_q(0)}\n  = 1+\\lim_{q\\to 0+}\\frac{h_q(x)}{r_q(0)}\n  =1.\n\\end{align}\n\nWe define \\(D = \\cbra{l\\ge 0\\colon \\eta_{l-} < \\eta_l}\\).\nThen the following formula is well-known in the excursion theory.\n\\begin{Lem}[{Compensation formula; see e.g.,\n  Bertoin~\\cite[Corollary IV.11]{MR1406564}}]\\label{Lem:compensation-formula}\n  Let \\(F(t,\\omega,e)\\) be a measurable functional\n  on \\([0, \\infty) \\times \\ensuremath{\\mathcal{D}} \\times \\ensuremath{\\mathcal{D}}\\) such that,\n  for every fixed \\(e\\in \\ensuremath{\\mathcal{D}}\\), the process\n  \\((F(t,\\cdot,e), t\\ge 0)\\) is \\((\\ensuremath{\\mathcal{F}}_t)\\)-predictable. Then\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra*{\\sum_{l\\in D} F(\\eta_{l-}, X, e_l)}\n    = \\ensuremath{\\mathbb{P}}\\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_t\\, F(t, X, \\widetilde{X})},\n  \\end{align}\n  where the symbol \\(\\widetilde{\\hspace{12pt}}\\) means independence.\n\\end{Lem}\n\nLet \\(L^a =\\rbra{L_t^a, t\\ge 0}\\) denote the local time at \\(a\\in \\ensuremath{\\mathbb{R}}\\)\nwhich is normalized by\n\\begin{align}\\label{eq:regularity-of-L^a}\n  \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt} \\mathrm{d} L^a_t} = r_q(a-x),\n  \\quad  q>0,\\;x \\in \\ensuremath{\\mathbb{R}}.\n\\end{align}\nWe denote by \\(\\eta^a=(\\eta^a_u, u\\ge 0)\\)\nthe right-continuous inverse of \\(L^a\\) given by\n\\(\\eta^a_u=\\inf\\cbra{t>0\\colon L_t^a>u}\\).\nWe denote by \\(n^a\\) the characteristic measure of excursions away from \\(a\\).\n\n\\section{The renormalized zero resolvent}\\label{Sec:h}\nLet us consider the existence and properties of\nthe renormalized zero resolvent in Theorems~\\ref{Thm:exist-h} and~\\ref{Thm:property-h}.\nRecall that we assume \\(X\\) is recurrent, i.e., \\(\\kappa=0\\),\nand assume the condition~\\ref{item:assumption}.\n\\subsection{Key lemmas for the renormalized zero resolvent}\nTo show Theorems~\\ref{Thm:exist-h} and~\\ref{Thm:property-h},\nwe prepare some lemmas.\nRecall that \\(m^2\\) has been introduced in~\\eqref{eq:second-moment};\n\\(m^2=\\ensuremath{\\mathbb{P}}\\sbra{X_1^2}\\).\n\\begin{Lem}\\label{Lem:psi-conv}\n  The following assertions hold.\n  \\begin{enumerate}\n    \\item If \\(m^2<\\infty\\), then\n          \\begin{align}\n            \\varPsi(\\lambda)\n            = \\frac{1}{2}\\sigma^2\\lambda^2\n            + \\int_\\ensuremath{\\mathbb{R}} \\rbra{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}+\\mathrm{i} \\lambda x}\\nu(\\mathrm{d} x),\n          \\end{align}\\label{Lem-item:psi-repre}\n          and\n          \\begin{align}\\label{eq:psi\/lambda2}\n            \\lim_{\\lambda \\to 0}\\frac{\\varPsi(\\lambda)}{\\lambda^2}\n            =\\lim_{\\lambda \\to 0}\\frac{\\theta(\\lambda)}{\\lambda^2}\n            = \\frac{m^2}{2}\n            =\\frac{1}{2}\\rbra*{\\sigma^2 + \\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x)}.\n          \\end{align}\\label{Lem-item:psi-conv}\n    \\item If \\(m^2=\\infty\\), then\n          \\begin{align}\n            \\lim_{\\lambda \\to 0}\\frac{\\lambda^2}{\\varPsi(\\lambda)}\n            =\\lim_{\\lambda \\to 0}\\frac{\\lambda^2}{\\theta(\\lambda)}=0.\n          \\end{align}\n  \\end{enumerate}\n\\end{Lem}\n\n\\begin{proof}\n  It is well-known\n  (see, e.g.,~\\cite[Theorem 2.7]{MR3155252})\n  that \\(X\\) has finite variance\n  if and only if \\(\\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x) < \\infty\\).\n  We first assume that\n  \\(\\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x) < \\infty\\). Then we know\n  \\(\\ensuremath{\\mathbb{P}}\\sbra{X_1}\\) and \\(\\ensuremath{\\mathbb{P}}\\sbra{X_1^2}\\) are finite and\n  \\begin{gather}\n    \\ensuremath{\\mathbb{P}}\\sbra{X_1}\n    =\\mathrm{i}\\varPsi^\\prime(0)\n    = - v + \\int_{\\ensuremath{\\mathbb{R}}\\setminus (-1, 1)} x \\nu(\\mathrm{d} x), \\\\\n    \\ensuremath{\\mathbb{P}}\\sbra{X_1^2}\n    = \\varPsi^{\\prime\\prime}(0)\n    = \\sigma^2 + \\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x) +\n    {\\ensuremath{\\mathbb{P}}\\sbra{X_1}}^2.\n  \\end{gather}\n  Since \\(X\\) is recurrent, we have \\(\\ensuremath{\\mathbb{P}}[X_1] = 0\\);\n  see, e.g.,~\\cite[Problem 7.2]{MR3155252}.\n  This implies that\n  \\begin{align}\n    \\varPsi(\\lambda)\n    = \\frac{1}{2} \\sigma^2 \\lambda\n    + \\int_\\ensuremath{\\mathbb{R}} \\rbra{1-\\mathrm{e}^{\\mathrm{i}\\lambda x} + \\mathrm{i} \\lambda x} \\nu(\\mathrm{d} x).\n  \\end{align}\n  By l'H\u00f4pital's rule, we obtain\n  \\begin{align}\n    \\lim_{\\lambda\\to 0} \\frac{\\varPsi(\\lambda)}{\\lambda^2}\n    = \\lim_{\\lambda\\to 0} \\frac{\\varPsi^\\prime(\\lambda)}{2\\lambda}\n    = \\frac{\\varPsi^{\\prime\\prime}(0)}{2} = \\frac{m^2}{2}.\n  \\end{align}\n  Taking real parts on both sides, we also have\n  \\begin{align}\n    \\lim_{\\lambda\\to 0}\\frac{\\theta(\\lambda)}{\\lambda^2} = \\frac{m^2}{2}.\n  \\end{align}\n  We next assume that \\(\\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x) = \\infty\\).\n  Then we know \\(m^2 = \\infty\\).\n  By~\\eqref{eq:theta}, we have\n  \\begin{align}\n    \\abs*{\\frac{\\varPsi(\\lambda)}{\\lambda^2}}\n    \\ge \\abs*{\\frac{\\theta(\\lambda)}{\\lambda^2}}\n    \\ge \\int_\\ensuremath{\\mathbb{R}} \\frac{1-\\cos \\lambda x}{\\lambda^2}\\nu(\\mathrm{d} x).\n  \\end{align}\n  Using Fatou's lemma and l'H\u00f4pital's rule, we obtain\n  \\begin{align}\n    \\liminf_{\\lambda\\to 0}\\abs*{\\frac{\\varPsi(\\lambda)}{\\lambda^2}}\n    \\ge \\liminf_{\\lambda\\to 0}\\abs*{\\frac{\\theta(\\lambda)}{\\lambda^2}}\n     & \\ge \\int_\\ensuremath{\\mathbb{R}} \\liminf_{\\lambda\\to 0}\n    \\frac{1-\\cos \\lambda x}{\\lambda^2}\n    \\nu(\\mathrm{d} x)                                       \\\\\n     & = \\int_\\ensuremath{\\mathbb{R}} \\frac{x^2}{2} \\nu(\\mathrm{d} x) = \\infty.\n  \\end{align}\n  Therefore the proof is complete.\n\\end{proof}\n\nWhen \\(m^2<\\infty\\), the next lemma\nis essential for the renormalized zero resolvent.\n\\begin{Lem}\\label{Lem:fin-omega-integral}\n  Assume \\(m^2 < \\infty\\).\n  Then it holds that\n  \\begin{align}\\label{eq:fin-omega-integral}\n    \\int_\\ensuremath{\\mathbb{R}} \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}} \\, \\mathrm{d} \\lambda \\\n    < \\infty.\n  \\end{align}\n  Consequently,\n  Tsukada's condition~\\ref{item:Tsukada-b} holds.\n\\end{Lem}\n\n\\begin{proof}[Proof of Lemma~\\ref{Lem:fin-omega-integral}]\n  By Lemma~\\ref{Lem:psi-conv},\n  we have\n  \\begin{align}\n    \\omega(\\lambda)\n    = \\int_{\\ensuremath{\\mathbb{R}}} \\rbra{\\lambda x - \\sin \\lambda x}\n    \\nu(\\mathrm{d} x).\n  \\end{align}\n  Hence we have\n  \\begin{align}\n    \\int_\\ensuremath{\\mathbb{R}} \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}} \\, \\mathrm{d} \\lambda\n     & = \\int_\\ensuremath{\\mathbb{R}} \\abs*{\\int_\\ensuremath{\\mathbb{R}}\n      \\frac{\\lambda x - \\sin \\lambda x}{\\lambda^3}\n      \\nu(\\mathrm{d} x)}\n    \\, \\mathrm{d} \\lambda                                  \\\\\n     & \\le \\rbra*{\\int_{-\\infty}^0 + \\int_0^\\infty}\n    \\rbra*{\\int_{-\\infty}^0 + \\int_0^\\infty}\n    \\abs*{\\frac{\\lambda x - \\sin \\lambda x}{\\lambda^3}}\n    \\nu(\\mathrm{d} x)\n    \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  Since \\(\\lambda x - \\sin \\lambda x \\ge 0\\)\n  for \\((x, \\lambda) \\in {(0, \\infty)}^2\\),\n  it holds that\n  \\begin{align}\n    \\int_0^\\infty \\int_0^\\infty\n    \\abs*{\\frac{\\lambda x - \\sin \\lambda x}{\\lambda^3}}\n    \\nu(\\mathrm{d} x)\n    \\, \\mathrm{d} \\lambda\n     & =\n    \\int_0^\\infty \\int_0^\\infty\n    \\frac{\\lambda x - \\sin \\lambda x}{\\lambda^3}\n    \\, \\mathrm{d} \\lambda \\,\n    \\nu(\\mathrm{d} x)                                                \\\\\n     & =\n    \\int_0^\\infty x^2 \\nu(\\mathrm{d} x)\n    \\int_0^\\infty\n    \\frac{\\xi - \\sin \\xi}{\\xi^3} \\, \\mathrm{d}\\xi                    \\\\\n     & = \\frac{4}{\\pi} \\int_0^\\infty x^2 \\nu(\\mathrm{d} x) < \\infty.\n  \\end{align}\n  Other integrals are also proved to be finite by the same discussions,\n  and we obtain~\\eqref{eq:fin-omega-integral}.\n  By~(\\ref{Lem-item:psi-repre}) of Lemma~\\ref{Lem:psi-conv},\n  we see that \\(\\lambda^2\/\\varPsi(\\lambda)\\) is bounded near \\(\\lambda=0\\).\n  Thus we have\n  \\begin{align}\n    \\int_0^1 \\abs*{\\Im\\rbra*{\\frac{\\lambda}{\\varPsi(\\lambda)}}}\\,\\mathrm{d}\\lambda\n    = \\int_0^1 \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}^2\n    \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}}\\, \\mathrm{d} \\lambda <\\infty.\n  \\end{align}\n  Since~\\ref{item:assumption} is assumed in this section,\n  this implies that Tsukada's condition~\\ref{item:Tsukada-b} holds.\n\\end{proof}\n\n\\begin{Lem}\\label{Lem:exist-hheq}\n  The following assertions hold.\n  \\begin{enumerate}\n    \\item \\(\\displaystyle\n          h^S(x)\n          \\coloneqq \\lim_{q\\to 0+}(h_q(x) + h_q(-x))\n          = \\frac{2}{\\pi}\n          \\int_0^\\infty\n          \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}} \\, \\mathrm{d} \\lambda,\n          \\quad\\text{for \\(x\\in\\ensuremath{\\mathbb{R}}\\).}\n          \\)\\label{Lem-item:exist-h^S}\n    \\item \\(\\displaystyle\n          \\lim_{x\\to\\pm\\infty} \\frac{h^S(x)}{\\abs{x}} =\n          \\frac{2}{m^2} \\in [0,\\infty).\n          \\)\\label{Lem-item:h^S\/x}\n    \\item For \\(x, y \\in \\ensuremath{\\mathbb{R}}\\),\n          \\begin{align}\n            h^D(x, y)\n             & \\coloneqq \\lim_{q\\to 0+}\\cbra{h_q(y+2x) - 2h_q(y+x) + h_q(y)} \\\\\n             & = \\frac{2}{\\pi} \\int_0^\\infty\n            \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i} \\lambda(y+x)}\n              \\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}} \\, \\mathrm{d} \\lambda.\n          \\end{align}\n          Moreover, it holds that\n          \\(\\lim_{y\\to\\pm\\infty} h^D(x, y) = 0\\).\\label{Lem-item:h^D}\n  \\end{enumerate}\n\\end{Lem}\n\n\\begin{proof}\n  \\noindent (\\ref{Lem-item:exist-h^S})\n  By~\\eqref{eq:def-h_q}, it holds that\n  \\begin{align}\\label{eq:h_q^s}\n    h_q(x)+h_q(-x)\n    = \\frac{2}{\\pi}\\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{q+\\varPsi(\\lambda)}}\n    \\,\\mathrm{d}\\lambda.\n  \\end{align}\n  Since \\(\\theta(\\lambda) \\ge 0\\), we have\n  \\(\\abs{q+\\varPsi(\\lambda)}\\ge \\abs{\\varPsi(\\lambda)}\\).\n  Hence it holds that\n  \\begin{align}\\label{eq:hs-conv-DCT}\n    \\abs*{\\Re\\rbra*{\\frac{1-\\cos \\lambda x}{q+\\varPsi(\\lambda)}}}\n    \\le \\abs*{\\frac{1-\\cos \\lambda x}{q+\\varPsi(\\lambda)}}\n    \\le \\frac{1-\\cos \\lambda x}{\\abs{\\varPsi(\\lambda)}}\n    \\le \\frac{{(\\lambda x)}^2 \\wedge 2}{\\abs{\\varPsi(\\lambda)}},\n  \\end{align}\n  which is integrable in \\(\\lambda>0\\) by Lemma~\\ref{Lem:inte-of-varPsi}.\n  Then we may apply the dominated convergence theorem\n  to deduce that\n  \\begin{align}\n    h_q(x)+h_q(-x) \\longrightarrow\n    \\frac{2}{\\pi}\\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\\,\\mathrm{d}\\lambda,\n    \\quad \\text{as \\(q \\to 0+\\).}\n  \\end{align}\n\n  \\noindent  (\\ref{Lem-item:h^S\/x})\n  We only consider the case \\(x \\to \\infty\\) since the case\n  \\(x \\to -\\infty\\) can be proved in the same way.\n  For any \\(\\delta>0\\), we have\n  \\begin{align}\n    \\abs*{\\frac{2}{\\pi x}\n    \\int_\\delta^\\infty\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda}\n    \\le \\frac{2}{\\pi\\abs{x}}\n    \\int_\\delta^\\infty \\abs*{\\frac{2}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    \\longrightarrow 0, \\quad \\text{as \\(x\\to\\infty\\).}\\label{eq:h^S\/x-big}\n  \\end{align}\n  Fix \\(\\varepsilon > 0\\).\n  By Lemma~\\ref{Lem:psi-conv},\n  we can choose \\(\\delta>0\\) such that\n  \\begin{align}\n    \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}\n      - \\frac{2}{m^2}} < \\varepsilon,\n    \\quad \\text{for } \\abs{\\lambda} < \\delta.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\frac{2}{\\pi x}\n    \\int_0^\\delta\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n     & \\le \\rbra*{\\frac{2}{m^2} + \\varepsilon}\n    \\frac{2}{\\pi x}\\int_0^\\delta\n    \\frac{1-\\cos \\lambda x}{\\lambda^2}\n    \\, \\mathrm{d} \\lambda                             \\\\\n     & \\to \\rbra*{\\frac{2}{m^2} + \\varepsilon}\n    \\frac{2}{\\pi}\n    \\int_0^\\infty \\frac{1-\\cos \\xi}{\\xi^2}\n    \\, \\mathrm{d} \\xi                                 \\\\\n     & = \\frac{2}{m^2} + \\varepsilon,\n    \\quad \\text{as \\(x\\to\\infty\\).}\n  \\end{align}\n  Thus we obtain\n  \\begin{align}\n    \\limsup_{x\\to\\infty}\n    \\frac{2}{\\pi x}\n    \\int_0^\\delta\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    \\le\n    \\frac{2}{m^2} + \\varepsilon.\\label{eq:h^S\/x-upper}\n  \\end{align}\n  In the same way, we can show that\n  \\begin{align}\n    \\liminf_{x\\to\\infty}\n    \\frac{2}{\\pi x}\n    \\int_0^\\delta\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    \\ge\n    \\frac{2}{m^2} - \\varepsilon.\\label{eq:h^S\/x-lower}\n  \\end{align}\n  By~\\eqref{eq:h^S\/x-big},~\\eqref{eq:h^S\/x-upper}\n  and~\\eqref{eq:h^S\/x-lower},\n  the result follows.\n\n  \\noindent (\\ref{Lem-item:h^D})\n  By~\\eqref{eq:def-h_q}, we have\n  \\begin{align}\n    h_q(y+2x) - 2h_q(y+x) + h_q(y)\n    =  \\frac{2}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{ \\mathrm{e}^{\\mathrm{i}\\lambda (y+x)}\n      \\frac{1-\\cos \\lambda x}{q+\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  By the same way as~\\eqref{eq:hs-conv-DCT},\n  we may apply the dominated convergence theorem\n  to obtain\n  \\begin{align}\n    h^D(x, y)\n     & = \\lim_{q\\to 0+} \\cbra{h_q(y+2x) - 2h_q(y+x) + h_q(y)} \\\\\n     & = \\frac{2}{\\pi} \\int_0^\\infty\n    \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i} \\lambda(y+x)}\n      \\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}} \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  Furthermore, by the Riemann--Lebesgue lemma, we obtain\n  \\(h^D(x, y) \\longrightarrow 0\\)\n  as \\(y \\to \\pm \\infty\\).\n\\end{proof}\n\n\\subsection{Proofs of Theorems~\\ref{Thm:exist-h}\n  and~\\ref{Thm:property-h}}\\label{Subsec:pf-h}\nWe separate the proof into the two cases:\n\\(m^2<\\infty\\) and \\(m^2=\\infty\\).\nWe first show the existence and properties of \\(h\\) in the case \\(m^2<\\infty\\).\nIn this case, we can use the dominated convergence theorem.\n\\begin{proof}[Proof of~(\\ref{Thm-item:exist-h}) of Theorem~\\ref{Thm:exist-h}\n    in the case \\(m^2<\\infty\\)]\n  For each \\(x, \\lambda \\in \\ensuremath{\\mathbb{R}}\\),\n  we observe that\n  \\begin{align}\n    \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{q+\\varPsi(\\lambda)}}\n    = \\frac{(q+\\theta(\\lambda)) (1-\\cos\\lambda x)\n      + \\omega(\\lambda) \\sin\\lambda x}\n    {\\abs{q+\\varPsi(\\lambda)}^2}.\n  \\end{align}\n  Hence it follows from \\(\\theta(\\lambda) \\ge 0\\) that\n  \\begin{align}\n     & \\abs*{\\Re\\rbra*{\n    \\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{q+\\varPsi(\\lambda)}}}    \\\\\n     & \\le\n    \\rbra*{\\frac{1-\\cos\\lambda x}{\\abs{\\varPsi(\\lambda)}}\n      + \\frac{\\lambda^4}\n      {\\abs{\\varPsi(\\lambda)}^2}\n      \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}}\n      \\abs*{\\frac{\\sin\\lambda x}{\\lambda}}}\n    \\wedge\n    \\abs*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{\\varPsi(\\lambda)}} \\\\\n     & \\le\n    \\rbra*{\\abs*{\\frac{\\lambda^2 x^2}{\\varPsi(\\lambda)}}\n      + \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}^2\n      \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}}\n      \\abs{x}}\n    \\wedge\n    \\abs*{\\frac{2}{\\varPsi(\\lambda)}}.\n  \\end{align}\n  By Lemma~\\ref{Lem:inte-of-varPsi},\n  (\\ref{Lem-item:psi-conv}) of Lemma~\\ref{Lem:psi-conv}\n  and Lemma~\\ref{Lem:fin-omega-integral},\n  the last quantity is integrable in \\(\\lambda>0\\).\n  Therefore, we may apply the dominated convergence theorem\n  to conclude that\n  \\begin{align}\n    h_q(x)\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{q+\\varPsi(\\lambda)}}\n    \\,\\mathrm{d} \\lambda \\\\\n     & \\to\n    \\frac{1}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{\\varPsi(\\lambda)}}\n    \\,\\mathrm{d} \\lambda,\n    \\quad \\text{as \\(q\\to 0+\\).}\n  \\end{align}\n  Hence the proof is complete.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{Thm:property-h}\n    in the case \\(m^2<\\infty\\)]\n  \\noindent (\\ref{Thm-item:h\/x-infinity})\n  We take \\(\\delta>0\\) sufficiently small.\n  By (\\ref{Thm-item:exist-h}) of Theorem~\\ref{Thm:exist-h},\n  we have\n  \\begin{align}\n    \\hspace{-20pt}\\frac{h(x)}{x}\n     & =\n    \\frac{1}{\\pi x}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i}\\lambda x}}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda \\\\\n     & =\n    \\frac{1}{\\pi x}\\cbra*{\n      \\int_\\delta^\\infty\n      \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i}\\lambda x}}{\\varPsi(\\lambda)}}\n      \\, \\mathrm{d} \\lambda\n      +\n      \\int_0^\\delta\n      \\frac{\\omega(\\lambda)\\sin\\lambda x}\n      {\\abs{\\varPsi(\\lambda)}^2}\n      \\, \\mathrm{d} \\lambda\n      +\n      \\int_0^\\delta\n      \\frac{\\theta(\\lambda)(1-\\cos\\lambda x)}\n      {\\abs{\\varPsi(\\lambda)}^2}\n      \\, \\mathrm{d} \\lambda}.\\label{eq:three-inte}\n  \\end{align}\n  For the first integral in~\\eqref{eq:three-inte}, we have\n  \\begin{align}\n    \\abs*{\n      \\frac{1}{\\pi x}\n      \\int_\\delta^\\infty\n      \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i}\\lambda x}}{\\varPsi(\\lambda)}}\n      \\, \\mathrm{d} \\lambda\n    }\n    \\le\n    \\frac{1}{\\pi\\abs{x}}\n    \\int_\\delta^\\infty\n    \\abs*{\\frac{2}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    \\longrightarrow 0,\n    \\quad \\text{as \\(x \\to \\pm\\infty\\).}\n  \\end{align}\n  For the second integral in~\\eqref{eq:three-inte}, since we have\n  \\begin{align}\\label{eq:sin-omega-DCT}\n    \\frac{\\abs{\\omega(\\lambda)\\sin\\lambda x}}\n    {\\abs{\\varPsi(\\lambda)}^2\\abs{x}}\n    \\le\n    \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}^2\n    \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}},\n  \\end{align}\n  which is integrable in \\(\\lambda\\in (0, \\delta)\\)\n  by (\\ref{Lem-item:psi-conv}) of Lemma~\\ref{Lem:psi-conv} and\n  Lemma~\\ref{Lem:fin-omega-integral},\n  we can apply the dominated convergence theorem to obtain\n  \\begin{align}\n    \\frac{1}{\\pi x}\n    \\int_0^\\delta\n    \\frac{\\omega(\\lambda)\\sin\\lambda x}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda\n    \\longrightarrow 0,\n    \\quad \\text{as \\( x \\to \\pm\\infty\\).}\n  \\end{align}\n  For the third integral in~\\eqref{eq:three-inte},\n  we can apply the similar discussion as\n  the proof of (\\ref{Lem-item:h^S\/x}) of Lemma~\\ref{Lem:exist-hheq}.\n  Therefore we obtain\n  \\begin{align}\n    \\lim_{x\\to\\pm\\infty} \\frac{h(x)}{\\abs{x}} = \\frac{1}{m^2}.\n  \\end{align}\n\n  \\noindent  (\\ref{Thm-item:h-h-limit})\n  Take \\(\\delta>0\\) sufficiently small.\n  Then we have\n  \\begin{align}\n     & h(y+x) - h(y) \\\\\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\n      \\frac{1-\\mathrm{e}^{\\mathrm{i}\\lambda x}}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda   \\\\\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\n      \\frac{1-\\cos\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    + \\frac{1}{\\pi}\n    \\rbra*{\\int_0^\\delta + \\int_\\delta^\\infty}\n    \\Im\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\\frac{\\sin\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  By the Riemann--Lebesgue lemma, we obtain\n  \\begin{align}\n    \\lim_{y\\to\\pm\\infty}\\int_0^\\infty\n    \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\n      \\frac{1-\\cos\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    =\\lim_{y\\to\\pm\\infty}\\int_\\delta^\\infty\n    \\Im\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\\frac{\\sin\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    =0.\n  \\end{align}\n  On the other hand, we have\n  \\begin{align}\n   \\hspace{-25pt}\\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\Im\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\\frac{\\sin\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\frac{\\theta(\\lambda)\\sin\\lambda x \\sin\\lambda y}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda\n    - \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\frac{\\omega(\\lambda)\\sin\\lambda x \\cos\\lambda y}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda.\\label{eq:h-property-finvar}\n  \\end{align}\n  By~\\eqref{eq:sin-omega-DCT},\n  we may apply the Riemann--Lebesgue lemma\n  to the second integral and we have\n  \\begin{align}\n    \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\frac{\\omega(\\lambda)\\sin\\lambda x \\cos\\lambda y}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda\n    \\longrightarrow 0,\n    \\quad \\text{as \\(y\\to\\pm\\infty\\).}\n  \\end{align}\n  For the first integral of~\\eqref{eq:h-property-finvar}, we use\n  Lemma~\\ref{Lem:psi-conv} and\n  Lemma~\\ref{Lem:JT-DI} below\n  to show that\n  \\begin{align}\n    \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\frac{\\theta(\\lambda)\\sin\\lambda x \\sin\\lambda y}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}^2\n    \\frac{\\theta(\\lambda)}{\\lambda^2}\n    \\frac{\\sin \\lambda x}{\\lambda}\n    \\frac{\\sin \\lambda y}{\\lambda}\n    \\, \\mathrm{d} \\lambda            \\\\\n     & \\to \\pm \\frac{x}{m^2},\n    \\quad \\text{as \\(y\\to\\pm\\infty\\).}\n  \\end{align}\n  This ends the proof.\n\\end{proof}\nThe following lemma is an elementary calculus.\n\\begin{Lem}[Jordan's theorem for the Dirichlet integral]\\label{Lem:JT-DI}\n  Let \\(\\delta>0\\) and let\n  \\(f\\colon (0,\\delta)\\to \\ensuremath{\\mathbb{R}}\\) be continuous and be of bounded variation.\n  Then it holds that\n  \\begin{align}\n    \\lim_{x\\to \\pm\\infty}\\frac{2}{\\pi} \\int_0^\\delta\n    f(\\lambda)\\frac{\\sin \\lambda x}{\\lambda} \\, \\mathrm{d} \\lambda=\\pm f(0+),\n  \\end{align}\n  where \\(f(0+)=\\lim_{\\lambda\\to 0+}f(\\lambda)\\).\n\\end{Lem}\n\\begin{proof}\n  By integration by parts, we have\n  \\begin{align}\n    \\frac{2}{\\pi} \\int_0^\\delta\n    f(\\lambda)\\frac{\\sin \\lambda x}{\\lambda} \\, \\mathrm{d} \\lambda\n    &= \\frac{2}{\\pi}  \\int_0^\\delta \\rbra*{\\int_0^\\lambda\n    \\mathrm{d} f( \\xi)}\\frac{\\sin \\lambda x}{\\lambda}\\, \\mathrm{d} \\lambda\n    +\\frac{2}{\\pi} f(0+) \\int_0^\\delta \\frac{\\sin \\lambda x}{\\lambda}\\, \\mathrm{d} \\lambda\\\\\n    &=\\frac{2}{\\pi}\\int_0^\\delta \\rbra*{\\int_\\xi^\\delta\n    \\frac{\\sin \\lambda x}{\\lambda}\\, \\mathrm{d} \\lambda} \\mathrm{d} f(\\xi)\n    +\\frac{2}{\\pi} f(0+) \\int_0^\\delta \\frac{\\sin \\lambda x}{\\lambda}\\, \\mathrm{d} \\lambda\\\\\n    &\\to 0 \\pm f(0+), \\qquad\\text{as \\(x\\to\\pm \\infty\\).}\n  \\end{align}\n  Thus we obtain the desired result.\n\\end{proof}\n\nThen let us prove the existence of \\(h\\)\nin the case \\(X\\) is recurrent\nand \\(m^2=\\infty\\). Its proof\nis quite different from that in the case \\(m^2<\\infty\\).\n\n\\begin{proof}[Proof of~(\\ref{Thm-item:exist-h}) of Theorem~\\ref{Thm:exist-h}\n    in the case \\(m^2=\\infty\\)]\n  Since \\(h_q(0) = 0\\),\n  we have the limit \\(h(0)=\\lim_{q\\to 0+}h_q(0)=0\\).\n\n  Fix \\(a \\ne 0\\) and set\n  \\begin{align}\n    \\overline{h}(a) = \\limsup_{q\\to 0+}h_q(a), \\quad\n    \\underline{h}(a) = \\liminf_{q\\to 0+}h_q(a).\n  \\end{align}\n  We also define\n  \\(\\varDelta = \\overline{h}(a) - \\underline{h}(a) \\ge 0\\) and\n  \\(A = \\{2^j a\\colon j=0,1,2,3,\\ldots\\}\\).\n\n  It follows from\n  \\(h_q(x) \\le h_q(x)+h_q(-x)\\)\n  and~\\eqref{eq:h_q^s}--\\eqref{eq:hs-conv-DCT}\n  that \\({\\{h_q(x)\\}}_{q>0}\\) is bounded for each \\(x\\in \\ensuremath{\\mathbb{R}}\\).\n  Hence, by the diagonal argument,\n  we can take two\n  sequences \\(\\{q_n\\}, \\{q^\\prime_n\\}\\),\n  which satisfies the following three conditions:\n  \\begin{itemize}\n    \\item \\( q_n, q^\\prime_n \\to 0+\\) as \\(n\\to\\infty\\);\n    \\item \\(\\lim_{n\\to\\infty}h_{q_n}(x)\\) and\n          \\(\\lim_{n\\to\\infty}h_{q^\\prime_n}(x)\\) exist and are finite for\n          each \\(x \\in A\\);\n    \\item \\( \\overline{h}(a) = \\lim_{n\\to\\infty} h_{q_n}(a)\\) and\n          \\(\\underline{h}(a) = \\lim_{n\\to\\infty} h_{q^\\prime_n}(a)\\).\n  \\end{itemize}\n  Then we define, for \\(x \\in A\\),\n  \\begin{align}\n    \\overline{h}(x) = \\lim_{n\\to\\infty}h_{q_n}(x),\n    \\quad\n    \\underline{h}(x) = \\lim_{n\\to\\infty} h_{q^\\prime_n}(x).\n  \\end{align}\n  By (\\ref{Lem-item:h^D}) of Lemma~\\ref{Lem:exist-hheq}, we have\n  \\begin{align}\n    h^D(x,0)=\\overline{h}(2x) - 2 \\overline{h}(x)\n    =\\underline{h}(2x) - 2 \\underline{h}(x),\n    \\quad x \\in A.\n  \\end{align}\n  This implies that\n  \\(\\overline{h}(2x) - \\underline{h}(2x)\n  =2(\\overline{h}(x) - \\underline{h}(x))\\).\n  Hence it holds that\n  \\(\\overline{h}(2^j a) - \\underline{h}(2^j a)\n  = 2^j \\varDelta\\), i.e.,\n  \\begin{align}\\label{eq:diff-overunder-h}\n    \\frac{\\overline{h}(2^j a)}{2^j a} -\n    \\frac{\\underline{h}(2^j a)}{2^j a}\n    = \\frac{\\varDelta}{a}, \\quad j = 0,1,2,3,\\ldots.\n  \\end{align}\n  It follows from (\\ref{Lem-item:h^S\/x}) of Lemma~\\ref{Lem:exist-hheq}\n  that\n  \\begin{align}\n    \\max\\cbra*{\\frac{\\overline{h}(2^j a)}{2^j a},\n      \\frac{\\underline{h}(2^j a)}{2^j a}}\n    \\le\n    \\frac{h^S(2^j a)}{2^j a} \\longrightarrow 0,\n    \\quad \\text{as \\(j\\to\\infty\\).}\n  \\end{align}\n  Thus, by letting \\(j\\to\\infty\\) in~\\eqref{eq:diff-overunder-h},\n  we obtain \\(\\varDelta = 0\\).\n  Therefore, we conclude that \\(h(a)=\\lim_{q\\to 0+}h_q(a)\\) exists.\n\\end{proof}\n\nNext, we prove (\\ref{Thm-item:unif-conv-h}) and (\\ref{Thm-item:subadditive-h})\nof Theorem~\\ref{Thm:exist-h}\nin both cases\n\\(m^2=\\infty\\) and \\(m^2<\\infty\\).\n\\begin{proof}[Proof of (\\ref{Thm-item:unif-conv-h}) and (\\ref{Thm-item:subadditive-h})\n    of Theorem~\\ref{Thm:exist-h}]\n  By the Markov property, we have, for \\(x,y\\in\\ensuremath{\\mathbb{R}}\\),\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_{x+y}\\sbra{\\mathrm{e}^{-qT_0}} =\\ensuremath{\\mathbb{P}}_0\\sbra{\\mathrm{e}^{-qT_{-x-y}}}\n    \\ge \\ensuremath{\\mathbb{P}}_0\\sbra{\\mathrm{e}^{-qT_{-x}} \\ensuremath{\\mathbb{P}}_{-x}\\sbra{\\mathrm{e}^{-qT_{-x-y}}}}\n    = \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_{0}}} \\ensuremath{\\mathbb{P}}_y\\sbra{\\mathrm{e}^{-qT_0}}.\n  \\end{align}\n  Since \\(h_q(x) =r_q(0)\\rbra{1-\\ensuremath{\\mathbb{P}}_x\\sbra{e^{-qT_0}}}\\)\n  and \\(\\rbra{1-\\ensuremath{\\mathbb{P}}_x\\sbra{e^{-qT_0}}}\\rbra{1-\\ensuremath{\\mathbb{P}}_y\\sbra{e^{-qT_0}}}\\ge 0\\),\n  it holds that\n  \\begin{align}\n    h_q(x+y) \\le h_q(x) +h_q(y).\n  \\end{align}\n  Hence \\(h_q\\) is subadditive.\n  (This proof can also be found in~\\cite[Lemma 3.3]{MR3689384}.)\n  Since \\(h_q\\) is non-negative and subadditive,\n  and by~\\eqref{eq:h_q^s} and~\\eqref{eq:hs-conv-DCT},\n  it holds that\n  \\begin{align}\n    \\abs{h_q(x+\\delta) - h_q(x)}\n    \\le h_q(\\delta) + h_q(-\\delta)\n    \\le \\int_0^\\infty\n    \\abs*{\\frac{\\rbra{\\lambda \\delta}^2 \\wedge 2}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  Hence \\({\\{h_q\\}}_{q>0}\\) is equi-continuous.\n  Equi-continuity and pointwise-convergence imply\n  the uniform convergence on compact subset of \\(\\ensuremath{\\mathbb{R}}\\).\n  The subadditivity of \\(h\\) follows directly from that of \\(h_q\\).\n\\end{proof}\nFinally, we show the properties of \\(h\\) in Theorem~\\ref{Thm:property-h}\nin the case \\(m^2=\\infty\\).\n\\begin{proof}[Proof of Theorem~\\ref{Thm:property-h}\n    in the case \\(m^2=\\infty\\)]\n  \\noindent (\\ref{Thm-item:h\/x-infinity})\n  This is directly from\n  (\\ref{Lem-item:h^S\/x}) of Lemma~\\ref{Lem:exist-hheq}.\n\n  \\noindent (\\ref{Thm-item:h-h-limit})\n  Since \\(h\\) is subadditive, we have\n  \\begin{align}\\label{eq:h-diff-sum}\n    \\sum_{k=1}^n \\cbra{h(kx+y)-h((k-1)x+y)}\n    = h(nx+y) - h(y) \\le h(nx).\n  \\end{align}\n  By (\\ref{Lem-item:h^D}) of Lemma~\\ref{Lem:exist-hheq},\n  it holds that\n  \\begin{align}\n    \\cbra{h(2x+y) - h(x+y)} - \\cbra{h(x+y) - h(y)}\n    \\longrightarrow 0,\n    \\quad \\text{as \\(y\\to\\pm\\infty\\).}\n  \\end{align}\n  Thus we have\n  \\begin{align}\\label{eq:h-diff-limsup}\n    \\limsup_{y\\to\\pm\\infty}\n    \\sum_{k=1}^n \\cbra{h(kx+y)-h((k-1)x+y)}\n    = n \\limsup_{y\\to\\pm\\infty} \\cbra{h(x+y) - h(y)}.\n  \\end{align}\n  Combining~\\eqref{eq:h-diff-sum} and~\\eqref{eq:h-diff-limsup}, we obtain\n  \\begin{align}\n    \\limsup_{y\\to\\pm\\infty} \\cbra{h(x+y) - h(y)}\n    \\le \\frac{h(nx)}{n}.\n  \\end{align}\n  Since we have \\(\\lim_{n\\to\\infty}\\frac{h(nx)}{n}=0\\)\n  by~(\\ref{Lem-item:h^D}) of Lemma~\\ref{Lem:exist-hheq}, we have\n  \\begin{align}\n    \\limsup_{y\\to\\pm\\infty} \\cbra{h(x+y) - h(y)} \\le 0.\n  \\end{align}\n  Replacing \\(x\\) with \\(-x\\), we also have\n  \\begin{align}\n    \\liminf_{y\\to\\pm\\infty} \\cbra{h(y) - h(y-x)} \\ge 0.\n  \\end{align}\n  Therefore we obtain\n  \\(\n  \\lim_{y\\to\\pm\\infty} \\cbra{h(x+y) - h(y)} = 0.\n  \\)\n\\end{proof}\n\n\\subsection{The function \\(h^B\\)}\nLet us compute \\(\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}\\) and \\(\\ensuremath{\\mathbb{P}}_x(T_a<T_b)\\).\n\\begin{Lem}\\label{Lem:h^B-hitting-h}\n  \\begin{enumerate}\n    \\item For \\(a \\in \\ensuremath{\\mathbb{R}}\\),\n          \\begin{align}\n            h^B_q(a) \\coloneqq\n            \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^{T_a} \\mathrm{e}^{-qt}\\, \\mathrm{d} L_t}\n            = h_q(a)+h_q(-a)-\\frac{h_q(a)h_q(-a)}{r_q(0)}.\n            \\label{eq:LT-hitting-expect-q}\n          \\end{align}\\label{Lem-item:LT-hitting-expect}\n          Consequently, it holds that\n          \\begin{align}\n            h^B(a) \\coloneqq \\lim_{q\\to 0+} h_q^B(a)\n            =\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}= h(a) + h(-a).\n            \\label{eq:h^B}\n          \\end{align}\\label{Lem-item:hq-two-hitting-time}\n    \\item For \\(x, a, b \\in \\ensuremath{\\mathbb{R}}\\), \\(a\\ne b\\) and\n          for \\(q>0\\), it holds that\n          \\begin{align}\n            \\begin{aligned}\n               & \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a < T_b}\n              \\\\\n               & =\n              \\frac{h_q(b-a)+h_q(x-b)-h_q(x-a)-h_q(x-b)h_q(b-a)\/r_q(0)}{h^B_q(a-b)}.\n            \\end{aligned}\n            \\label{eq:qT_a;T_a<T_b}\n          \\end{align}\n          Consequently, it holds that\n          \\begin{align}\n            \\ensuremath{\\mathbb{P}}_x(T_a < T_b)\n            = \\frac{h(b-a)+h(x-b)-h(x-a)}{h^B(a-b)}.\n            \\label{eq:two-hittig-time-prob}\n          \\end{align}\\label{Lem-item:two-hittig-time-prob}\n  \\end{enumerate}\n\\end{Lem}\n\n\\begin{proof}\n  \\noindent (\\ref{Lem-item:hq-two-hitting-time})\n  We omit the proof of~\\eqref{eq:LT-hitting-expect-q},\n  which can be found\n  in~\\cite[Lemma V.11]{MR1406564}.\n  Letting \\(q\\to 0+\\) in~\\eqref{eq:LT-hitting-expect-q},\n  and using~\\eqref{eq:kappa},\n  we obtain~\\eqref{eq:h^B}.\n\n  \\noindent (\\ref{Lem-item:two-hittig-time-prob})\n  By the strong Markov property, it holds that,\n  for \\(x,a,b \\in \\ensuremath{\\mathbb{R}},\\,a\\ne b, \\, q>0\\),\n  \\begin{gather}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}}\n    = \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a};T_a < T_b}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_b}; T_b < T_a}\\ensuremath{\\mathbb{P}}_b\\sbra{\\mathrm{e}^{-qT_a}}, \\\\\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_b}}\n    = \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_b};T_b < T_a}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a < T_b}\\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-qT_b}}.\n  \\end{gather}\n  Combining the above two equalities, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}\n    = \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}}-\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_b}}\n      \\ensuremath{\\mathbb{P}}_b\\sbra{\\mathrm{e}^{-qT_a}}}\n    {1-\\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-qT_b}}\\ensuremath{\\mathbb{P}}_b\\sbra{\\mathrm{e}^{-qT_a}}}.\n  \\end{align}\n  By~\\eqref{eq:exp-hit-time}, this implies that\n  \\begin{align}\n     & \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}\n    \\\\\n     & = \\frac{r_q(a-x) - r_q(b-x)r_q(a-b)\/r_q(0)}\n    {r_q(0) - r_q(b-a)r_q(a-b)\/r_q(0)}\n    \\\\\n     & = \\frac{h_q(b-a)+h_q(x-b)-h_q(x-a)-h_q(x-b)h_q(b-a)\/r_q(0)}{h^B_q(a-b)}.\n  \\end{align}\n  Hence we obtain~\\eqref{eq:qT_a;T_a<T_b}.\n  Letting \\(q\\to 0+\\) in~\\eqref{eq:qT_a;T_a<T_b},\n  we obtain~\\eqref{eq:two-hittig-time-prob}.\n\\end{proof}\n\\begin{Rem}\n  The formulae~\\eqref{eq:qT_a;T_a<T_b} and~\\eqref{eq:two-hittig-time-prob} are\n  also discussed in Theorem 6.5\n  of Getoor~\\cite{MR185663}.\n  See also Proposition 5.3, Proposition 5.4\n  and Remark 5.5 of Yano--Yano--Yor~\\cite{MR2599211}.\n\\end{Rem}\nThe next theorem can be proved in the same way as\nPant\\'{\\i}~\\cite[Lemma 3.10]{MR3689384}.\n\n\\begin{Lem}[{\\cite[Lemma 3.10]{MR3689384}}]\\label{Thm:h^B-infty}\n  It holds that \\(\\lim_{x\\to\\infty} h^B(x) = \\infty\\).\n\\end{Lem}\n\\begin{proof}\n  For completeness of this paper, we give the proof.\n  Let \\(\\bm{e}\\) be an independent exponential time of mean \\(1\\)\n  and set \\(\\bm{e}_q \\coloneqq \\bm{e}\/q\\) for \\(q>0\\).\n  Then we have\n  \\begin{align}\\label{eq:h_b-eq}\n    h^B(x) =\\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}}\n    = \\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}; T_x\\le \\bm{e}_q} + \\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}; T_x > \\bm{e}_q}\n    \\ge \\ensuremath{\\mathbb{P}}\\sbra{L_{\\bm{e}_q}; T_x>\\bm{e}_q}.\n  \\end{align}\n  Letting \\(x\\to\\infty\\), we have\n  \\begin{align}\\label{eq:h_b-lower}\n    \\liminf_{x\\to\\infty}h^B(x) \\ge \\ensuremath{\\mathbb{P}}\\sbra{L_{\\bm{e}_q}} = r_q(0),\n    \\quad \\text{for all \\(q>0\\).}\n  \\end{align}\n  Since \\(X\\) is recurrent, i.e.,\n  \\(\\kappa=0\\) in~\\eqref{eq:kappa},\n  we let \\(q\\to 0+\\) to obtain\n  \\( \\liminf_{x\\to\\infty}h^B(x) \\ge \\infty\\).\n  Hence we obtain the desired result.\n\\end{proof}\n\nThe following theorem, which will be used in Section~\\ref{Sec:inv-time},\nis a generalization of the result\nin the symmetric case\nby Yano~\\cite[Theorem 6.1]{MR2603019}.\n\\begin{Thm}\\label{Thm:n-two-hitting-time}\n  For \\(a \\in \\ensuremath{\\mathbb{R}} \\setminus \\{0\\}\\), it holds that\n  \\begin{align}\\label{eq:n-T_a-T_0-inf}\n    n(T_a < T_0) = \\frac{1}{h^B(a)}.\n  \\end{align}\n\\end{Thm}\n\\begin{proof}\n  For \\(l > 0\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}(L_{T_a}>l) = \\ensuremath{\\mathbb{P}}(T_a > \\eta_l) =\n    \\ensuremath{\\mathbb{P}}(\\sigma_{\\cbra{T_a<T_0}}> l),\n  \\end{align}\n  where \\(\\sigma_A = \\inf\\cbra{l \\colon e_l \\in A}\\)\n  for \\(A \\subset \\ensuremath{\\mathcal{D}}\\).\n  Since \\(\\sigma_A\\) is the hitting time\n  of the set \\(A\\)\n  for the\n  Poisson point process \\(((l, e_l), l\\ge 0)\\),\n  we have\n  \\begin{align}\\label{eq:L_T_a-exp-distrib}\n    \\ensuremath{\\mathbb{P}}(L_{T_a}>l)\n    = \\mathrm{e}^{-l{n(T_a<T_0)}}.\n  \\end{align}\n  For more details, see, e.g.,~\\cite[Lemma 6.17]{MR3155252}.\n  In particular, \\(L_{T_a}\\) is exponentially distributed.\n  On the other hand, we know \\(h^B(a) = \\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}\\)\n  by Lemma~\\ref{Lem:h^B-hitting-h}.\n  Hence we obtain~\\eqref{eq:n-T_a-T_0-inf}.\n\\end{proof}\n\n\\subsection{Examples of the renormalized zero resolvent}\n\\begin{Eg}[Brownian motion]\\label{Eg:h-brownian}\n  Assume \\(X\\) is a standard Brownian motion.\n  Since \\(m^2=\\sigma^2=1\\), it holds that\n  \\begin{align}\n    h^{(\\gamma)}(x) = \\abs{x}+\\gamma x=\\begin{cases}\n      (1+\\gamma)x & x\\ge 0, \\\\\n      (1-\\gamma)x & x\\le 0,\n    \\end{cases}\n    \\quad\\text{for \\(-1\\le \\gamma\\le 1\\).}\n  \\end{align}\n\\end{Eg}\n\\begin{Eg}[Strictly stable process]\n  Assume that \\(X\\) is a strictly stable process of index\n  \\(\\alpha\\in (1,2)\\) with the L\\'{e}vy measure\n  \\begin{align}\n    \\nu(\\mathrm{d} x) =\n    \\begin{cases}\n      c_{+} \\abs{x}^{-\\alpha-1}\\, \\mathrm{d} x & \\text{on \\((0, \\infty)\\),}  \\\\\n      c_{-} \\abs{x}^{-\\alpha-1}\\, \\mathrm{d} x & \\text{on \\((-\\infty, 0)\\),}\n    \\end{cases}\n  \\end{align}\n  where \\(c_{+},c_{-} \\ge 0\\) and \\(c_{+}+c_{-} > 0\\).\n  Then its characteristic exponent is given by\n  \\begin{align}\n    \\varPsi(\\lambda)\n    = c\\abs{\\lambda}^{\\alpha}\n    \\rbra*{1-\\mathrm{i} \\beta \\sgn(\\lambda)\\tan\\frac{\\alpha\\pi}{2}},\n  \\end{align}\n  where \\(c\\) and \\(\\beta\\) are constants defined by\n  \\begin{align}\n    c = \\frac{(c_{+}+c_{-})\\pi}{2\\alpha \\Gamma(\\alpha)\\sin(\\pi\\alpha\/2)},\n    \\quad\n    \\beta = \\frac{c_{+} - c_{-}}{c_{+}+c_{-}}.\n  \\end{align}\n  In this case, we have \\(m^2=\\infty\\) and the function\n  \\(h\\) can be represented as\n  \\begin{align}\\label{eq:stable-h}\n    h(x) = \\frac{1}{K(\\alpha)} (1-\\beta\\sgn (x))\\abs{x}^{\\alpha-1},\n  \\end{align}\n  where\n  \\begin{align}\n    K(\\alpha) = -2c\\Gamma(\\alpha)\\cos\\frac{\\pi\\alpha}{2}\n    \\rbra*{1+\\beta^2\\tan^2 \\frac{\\pi\\alpha}{2}}.\n  \\end{align}\n  For more details, see~\\cite[Section 5]{MR3072331}.\n\\end{Eg}\n\n\\section{Local time penalization with exponential clock}\\label{Sec:exp-clock}\nWe now start to deal with the penalization\nresult with exponential clock.\nLet \\(\\bm{e}\\) be an independent exponential time of mean\n\\(1\\) and for \\(q>0\\), we write \\(\\bm{e}_q \\coloneqq \\bm{e}\/q\\), which\nhas an exponential distribution of mean \\(1\/q\\).\nWe compute \\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\\),\n\\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})|\\ensuremath{\\mathcal{F}}_t}\\) and its limit as \\(q\\to 0+\\) to\ninvestigate \\(\\lim_{q\\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{F_t\n  f(L_{\\bm{e}_q})}\/\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\\) for\nbounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functional \\(F_t\\).\nRecall that we assume \\(X\\) is recurrent\nand assume the condition~\\ref{item:assumption}.\n\\subsection{The law of the local time with exponential clock}\nFirst, we compute \\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\\).\n\\begin{Lem}\\label{Lem:expect-L_e_q}\n  Let \\(f\\) be a non-negative measurable function.\n  Then, for \\(q>0\\) and \\(x \\in \\mathbb{R}\\), it holds that\n  \\begin{align}\\label{eq:lem-expect-L_e_q}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\n    = \\frac{1}{r_q(0)}\\cbra*{h_q(x)f(0)\n    + \\rbra*{1-\\frac{h_q(x)}{r_q(0)}}\n    \\int_0^\\infty \\mathrm{e}^{-u\/{r_q(0)}}f(u) \\, \\mathrm{d} u}.\n  \\end{align}\n\\end{Lem}\n\\begin{proof}\n  Using the excursion theory, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_0\\sbra*{\\int_0^\\infty f(L_t) q \\mathrm{e}^{-qt}\\, \\mathrm{d} t}\n     & = \\ensuremath{\\mathbb{P}}_0\\sbra*{\\sum_{u\\in D}\n    \\int_{\\eta_{u-}}^{\\eta_u} f(u)q\\mathrm{e}^{-qu}\\, \\mathrm{d} u} \\\\\n     & = \\ensuremath{\\mathbb{P}}_0 \\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_t \\, f(L_t) \\mathrm{e}^{-qt}\n      \\int_0^{\\widetilde{T}_0} \\mathrm{d} u \\, q\\mathrm{e}^{-qu}},\n  \\end{align}\n  where the last equality follows from Lemma~\\ref{Lem:compensation-formula}.\n  By~\\eqref{eq:exp-nt0} and~\\eqref{eq:rel-n-r_q}, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_0 \\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_t \\, f(L_t) \\mathrm{e}^{-qt}\n      \\int_0^{\\widetilde{T}_0} \\mathrm{d} u \\, q\\mathrm{e}^{-qu}}\n     & = \\ensuremath{\\mathbb{P}}_0\\sbra*{\\int_0^\\infty \\mathrm{d} u \\, f(u) \\mathrm{e}^{-q\\eta_u}}\n    n\\sbra*{1 - \\mathrm{e}^{-qT_0}}                                            \\\\\n     & = \\frac{1}{r_q(0)}\\int_0^\\infty f(u) \\mathrm{e}^{-u\/{r_q(0)}} \\, \\mathrm{d} u.\n  \\end{align}\n  Applying the Markov property, we obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\n     & = \\ensuremath{\\mathbb{P}}_x \\sbra*{\\int_0^\\infty f(L_t)q\\mathrm{e}^{-qt}\\, \\mathrm{d} t}                    \\\\\n     & = \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^{T_0} f(0) q\\mathrm{e}^{-qt}\\, \\mathrm{d} t}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_0}}\\ensuremath{\\mathbb{P}}_0\\sbra*{\\int_0^\\infty f(L_t)q\\mathrm{e}^{-qt}\\, \\mathrm{d} t} \\\\\n     & = f(0)\\rbra*{1-\\frac{r_q(-x)}{r_q(0)}}\n    + \\frac{r_q(-x)}{r_q(0)} \\int_0^\\infty f(u) \\mathrm{e}^{-u\/{r_q(0)}}\\, \\mathrm{d} u        \\\\\n     & = \\frac{1}{r_q(0)}\\cbra*{h_q(x)f(0)\n    + \\rbra*{1-\\frac{h_q(x)}{r_q(0)}}\\int_0^\\infty \\mathrm{e}^{-u\/{r_q(0)}}f(u) \\, \\mathrm{d} u},\n  \\end{align}\n  here we used~\\eqref{eq:exp-hit-time}.\n  Therefore we obtain the desired result.\n\\end{proof}\n\n\\subsection{A.s.\\ convergence for exponential clock}\nTo calculate \\(\\ensuremath{\\mathbb{P}}_x\\sbra{F(L_{\\bm{e}_q})|\\ensuremath{\\mathcal{F}}_t}\\),\nwe separate into the two cases \\(\\cbra{t<\\bm{e}_q }\\) and \\(\\cbra{\\bm{e}_q\\le t}\\).\n\nLet \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\) and \\(x \\in \\ensuremath{\\mathbb{R}}\\). For \\(q > 0\\),\ndefine\n\\begin{align}\n  N_t^{q} & = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}); t < \\bm{e}_q | \\ensuremath{\\mathcal{F}}_t}, \\\\\n  M_t^{q} & = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}) | \\ensuremath{\\mathcal{F}}_t},               \\\\\n  A_t^{q} & = M_t^{q} - N_t^{q}\n  = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}); \\bm{e}_q \\le t | \\ensuremath{\\mathcal{F}}_t},\n\\end{align}\nand \\(h^{(\\gamma)}\\) and \\(M_t^{(\\gamma)}\\) are defined in~\\eqref{eq:def-h-gamma}\nand~\\eqref{eq:def-mart}.\n\n\\begin{Thm}\n  For \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\) and \\(x \\in \\ensuremath{\\mathbb{R}}\\), it holds that\n  \\begin{align}\n    \\lim_{q\\to 0+}N_t^{q}\n    =\\lim_{q\\to 0+} M_t^{q}\n    = M_t^{(0)}, \\qquad \\ensuremath{\\mathbb{P}}_x\\text{-a.s.}\n  \\end{align}\n\\end{Thm}\n\\begin{proof}\n  By the Markov property and the additivity of \\(L\\),\n  we have\n  \\begin{align}\n    N_t^{q}\n     & = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}); t < \\bm{e}_q | \\ensuremath{\\mathcal{F}}_t}\n    \\\\\n     & = r_q(0) \\mathrm{e}^{-qt} \\widetilde{\\ensuremath{\\mathbb{P}}}_{X_t}\n    \\sbra{f(L_t+\\widetilde{L}_{\\widetilde{\\bm{e}}_q})}\n    \\\\\n     & = \\mathrm{e}^{-qt}\n    \\cbra*{h_q(X_t)f(L_t) +\n      \\rbra*{1-\\frac{h_q(X_t)}{r_q(0)}}\n      \\int_0^\\infty \\mathrm{e}^{-u\/r_q(0)} f(L_t + u) \\, \\mathrm{d} u},\n  \\end{align}\n  here the last equality, we used Lemma~\\ref{Lem:expect-L_e_q}.\n  Since \\(1-\\frac{h_q(X_t)}{r_q(0)}=\\ensuremath{\\mathbb{P}}_{X_t}\\sbra{\\mathrm{e}^{-qT_0}} \\to 1\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q\\to 0+\\) and\n  since \\(\\int_0^\\infty f(L_t+u)\\, \\mathrm{d} u<\\infty\\),\n  we may apply the dominated convergence theorem to deduce that\n  \\(N_t^{q} \\longrightarrow M_t^{(0)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q\\to 0+\\).\n  By (\\ref{Lem-item:qr_q}) of Lemma~\\ref{Lem:inte-of-varPsi}, we have\n  \\begin{align}\n    A_t^{q} & = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}); \\bm{e}_q \\le t | \\ensuremath{\\mathcal{F}}_t}\n    = qr_q(0) \\int_0^t f(L_u) \\mathrm{e}^{-qu} \\, \\mathrm{d} u\n    \\to 0\n  \\end{align}\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q \\to 0+\\).\n  Therefore, we obtain \\(M_t^{q} \\to M_t^{(0)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q \\to 0+\\).\n\\end{proof}\n\n\\subsection{\\(\\ensuremath{\\mathcal{L}}^1\\) convergence for exponential clock}\nNow we prepare some lemma to prove the \\(\\ensuremath{\\mathcal{L}}^1\\)\nconvergence for exponential clock.\nThe following lemma is a part of\nTheorem 15.2 of Tsukada~\\cite{MR3838874}.\n\\begin{Lem}[{\\cite[Theorem 15.2]{MR3838874}}]\\label{Lem:h-l1-conv}\n  For \\(t \\ge 0\\), it holds that\n  \\(\n  h_q(X_t) \\longrightarrow h(X_t)\n  \\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(q \\to 0+\\).\n\\end{Lem}\nThe next theorem is the penalization result with exponential clock.\n\\begin{Thm}\\label{Thm:exp-L^1-conv}\\label{Thm:exp-time-result}\n  Let \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\) and \\(x \\in \\ensuremath{\\mathbb{R}}\\). Then\n  \\(( M_t^{(0)}, t \\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale,\n  and it holds that\n  \\begin{align}\n    \\lim_{q\\to 0+}N_t^{q}\n    = \\lim_{q\\to 0+}M_t^{q}\n    = M_t^{(0)},\n    \\qquad \\text{in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n  Consequently, if \\(M_0^{(0)}>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\\label{eq:exp-penalized-meas}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\bm{e}_q})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t^{(0)}}{M_0^{(0)}}},\n    \\qquad \\text{as \\(q \\to 0+\\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\nNote that the penalized measure in~\\eqref{eq:exp-penalized-meas}\nis not the same as that of Theorems~\\ref{Thm:hitting-time-result}\nand~\\ref{Thm:two-point-hitting-time-result}.\n\\begin{proof}[Proof of Theorem~\\ref{Thm:exp-L^1-conv}]\n  We first consider the case where \\(f\\) is bounded.\n  We write\n  \\begin{align}\n    N_t^{q}\n     & = \\mathrm{e}^{-qt}\\cbra*{h_q(X_t)f(L_t) +\n      \\rbra*{1-\\frac{h_q(X_t)}{r_q(0)}}\n    \\int_0^\\infty \\mathrm{e}^{-u\/r_q(0)} f(L_t + u) \\, \\mathrm{d} u} \\\\\n     & \\eqqcolon {(\\mathrm{I})}_q + {(\\mathrm{II})}_q, \\\\\n    M_t^{(0)}\n     & = h(X_t)f(L_t) +\n    \\int_0^\\infty\n    f(L_t + u) \\, \\mathrm{d} u                                \\\\\n     & \\eqqcolon {(\\mathrm{I})} + {(\\mathrm{II})}.\n  \\end{align}\n  By Lemma~\\ref{Lem:h-l1-conv}\n  and by the boundedness of \\(f\\),\n  we obtain \\({(\\mathrm{I})}_q \\to (\\mathrm{I})\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  Moreover, since \\(\\int_0^\\infty f(u)\\, \\mathrm{d} u<\\infty\\),\n  it follows from the dominated convergence theorem that\n  \\({(\\mathrm{II})}_q \\to (\\mathrm{II})\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  Hence we obtain \\(N_t^{q} \\to M_t^{(0)}\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  By (\\ref{Lem-item:qr_q}) of Lemma~\\ref{Lem:inte-of-varPsi},\n  we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{A_t^{q}}\n    = q r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^t \\mathrm{e}^{-qu} f(L_u)\\ \\mathrm{d} u}\n    \\le q r_q(0) t\\norm{f}\n    \\to 0, \\qquad \\text{as } q \\to {0+}.\n  \\end{align}\n  Since \\(A_t^q\\ge 0\\),\n  this means that \\(A_t^q\\to 0\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\). Thus we have\n  \\(M_t^{q} \\to M_t^{(0)}\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  For \\(0 \\le s \\le t\\), we know \\(\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{q}|\\ensuremath{\\mathcal{F}}_s} = M_s^{q}\\).\n  Letting \\(q \\to 0+\\) on both sides, we have\n  \\begin{align}\\label{eq:exp-mart}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(0)}|\\ensuremath{\\mathcal{F}}_s} = M_s^{(0)},\n  \\end{align}\n  which means that \\((M_t^{(0)},t\\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\\\n\n  Let us consider the general \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\).\n  We know the equality~\\eqref{eq:exp-mart} holds for \\(f \\wedge n\\).\n  Letting \\(n \\to \\infty\\),~\\eqref{eq:exp-mart}\n  holds for general \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\)\n  by the monotone convergence theorem.\n  Hence we have\n  \\begin{align}\n    \\lim_{q \\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{q}} = \\lim_{q \\to 0+}M_0^{q}\n    = M_0^{(0)} = \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(0)}},\n  \\end{align}\n  and by Fatou's lemma,\n  \\begin{align}\n    M_0^{(0)}\n    = \\lim_{q \\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{q}}\n    \\ge \\limsup_{q \\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{q}}\n    \\ge \\liminf_{q \\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{q}}\n    \\ge M_0^{(0)}.\n  \\end{align}\n  Thus we have\n  \\(\\lim_{q\\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{q}}=\\lim_{q\\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{q}}\n  =\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(0)}}\\).\n  Applying Scheff\\'{e}'s lemma, we obtain\n  \\(\\lim_{q\\to 0+}N_t^{q}=\\lim_{q\\to 0+} M_t^{q} = M^{(0)}_t\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n\\end{proof}\n\n\\section{Local time penalization with hitting time clock}\\label{Sec:hitting-time}\nWe deal with the penalization result with hitting time clock \\((T_a)\\).\nTo this aim, we compute \\(\\ensuremath{\\mathbb{P}}_x\\sbra{L_{T_a}}\\)\nand \\(\\ensuremath{\\mathbb{P}}_x\\sbra{L_{T_a}|\\ensuremath{\\mathcal{F}}_t}\\) and its limit as \\(a\\to\\pm\\infty\\).\nRecall that we assume \\(X\\) is recurrent\nand assume the condition~\\ref{item:assumption}.\n\\subsection{The law of the local time with hitting time clock}\nFirst, we compute \\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a})}\\).\n\\begin{Lem}\\label{Lem:expect-L_T_a}\n  For \\(x, a \\in \\ensuremath{\\mathbb{R}}\\), \\(a\\ne 0\\) and any non-negative measurable\n  function \\(f\\), we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a})}\n    = \\ensuremath{\\mathbb{P}}_x(T_0 > T_a) f(0)\n    + \\frac{\\ensuremath{\\mathbb{P}}_x(T_0<T_a) }{h^B(a)}\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)} f(u) \\, \\mathrm{d} u.\n  \\end{align}\n\\end{Lem}\n\\begin{proof}\n  By~\\eqref{eq:L_T_a-exp-distrib},\n  \\(L_{T_a}\\) is exponentially distributed with\n  parameter \\(1\/h^B(a)\\).\n  Hence we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_0\\sbra{f(L_{T_a})}\n    = \\frac{1}{h^B(a)}\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)} f(u) \\, \\mathrm{d} u.\n  \\end{align}\n  Applying the Markov property,\n  we conclude that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a})}\n     & = \\ensuremath{\\mathbb{P}}_x( T_0 > T_a) f(0) + \\ensuremath{\\mathbb{P}}_x(T_0 < T_a) \\ensuremath{\\mathbb{P}}_0\\sbra{f(L_{T_a})} \\\\\n     & = \\ensuremath{\\mathbb{P}}_x(T_0 > T_a)f(0)\n    + \\frac{\\ensuremath{\\mathbb{P}}_x(T_0<T_a) }{h^B(a)}\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)} f(u) \\, \\mathrm{d} u.\n  \\end{align}\n  Hence the proof is complete.\n\\end{proof}\n\n\\subsection{Proof of a.s.\\ convergence of\n  Theorem~\\ref{Thm:hitting-time-result}}\\label{Subsec:pf-hitting}\nWe now proceed the proof of the hitting time result.\nWe separate into the two cases \\(\\cbra{t<T_a}\\) and \\(\\cbra{T_a\\le t}\\).\n\\begin{proof}[Proof of a.s.\\ convergence of Theorem~\\ref{Thm:hitting-time-result}]\n  By the strong Markov property, the additivity of \\(L\\)\n  and Lemma~\\ref{Lem:expect-L_T_a}, we have\n  \\begin{align}\n    N_t^a\n     & =\n    1_{\\cbra{t<T_a}} h^B(a)\n    \\widetilde{\\ensuremath{\\mathbb{P}}}_{X_t}\n    \\sbra{f(\\widetilde{L}_{\\widetilde{T}_a}+L_t)} \\\\\n     & =\n    1_{\\cbra{t<T_a}}\n    \\cbra*{h^B(a)\\ensuremath{\\mathbb{P}}_{X_t}(T_0 > T_a)f(L_t)\n      + \\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_a)\n      \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)}f(L_t+u)\n      \\, \\mathrm{d} u\n    },\n  \\end{align}\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(a\\to\\pm\\infty\\).\n  (\\ref{Thm-item:h-h-limit}) of Theorem~\\ref{Thm:property-h}\n  and~\\eqref{eq:two-hittig-time-prob} imply that\n  \\begin{gather}\n    h^B(a)\\ensuremath{\\mathbb{P}}_{X_t}(T_0> T_a)\n    = h(X_t)+h(-a)-h(X_t-a)\n    \\longrightarrow h^{(\\pm 1)}(X_t) , \\\\\n    \\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_a) \\longrightarrow 1,\n  \\end{gather}\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(a\\to\\pm\\infty\\).\n  By Lemma~\\ref{Thm:h^B-infty}, we obtain\n  \\(N_t^a \\to M_t^{(\\pm 1)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(a\\to\\pm\\infty\\).\n  Furthermore, we have\n  \\begin{align}\n    M_t^a - N_t^a\n     & =\n    h^B(a) \\ensuremath{\\mathbb{P}}_x\\sbra*{f(L_{T_a}); T_a\\le t|\\ensuremath{\\mathcal{F}}_t}\\label{eq:hitting-A} \\\\\n     & =\n    h^B(a) f(L_{T_a}) 1_{\\cbra{T_a\\le t}}                              \\\\\n     & \\to 0,\n    \\quad \\ensuremath{\\mathbb{P}}_x\\text{-a.s.\\ as \\(a\\to\\pm\\infty\\).}\\label{eq:hitting-Al}\n  \\end{align}\n  Hence we obtain \\(M_t^a \\to M_t^{(\\pm 1)}\\)\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(a\\to\\pm\\infty\\).\n\\end{proof}\n\n\\subsection{Proof of \\(\\ensuremath{\\mathcal{L}}^1\\) convergence\n  of Theorem~\\ref{Thm:hitting-time-result}}\\label{Subsec:pf-hitting-l1}\n\\begin{proof}[Proof of \\(\\ensuremath{\\mathcal{L}}^1\\) convergence of Theorem~\\ref{Thm:hitting-time-result}]\n  We first consider the case\n  \\(m^2 = \\infty\\).\n  Then we know \\(M_t^{(\\pm 1)} = M_t^{(0)}\\).\n  Hence, by Theorem~\\ref{Thm:exp-L^1-conv},\n  \\(( M_t^{(\\pm 1)}, t \\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  Thus we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{a}} = M_0^{a}\n    \\to M_0^{(\\pm 1)} = \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}},\n    \\quad \\text{as \\(a\\to\\pm\\infty\\).}\n  \\end{align}\n  By Fatou's lemma, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}}\n    =\\lim_{a\\to\\pm\\infty}\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{a}}\n    \\ge \\limsup_{a\\to\\pm\\infty}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{a}}\n    \\ge \\liminf_{a\\to\\pm\\infty}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{a}}\n    \\ge \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}}.\n  \\end{align}\n  Consequently, it holds that\n  \\(\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{a}}, \\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{a}}\n  \\to \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}}\\), as \\(a\\to\\pm\\infty\\).\n  Applying Scheff\\'{e}'s lemma,\n  we obtain\n  \\(N_t^{a}, N_t^{a} \\to M_t^{(\\pm 1)}\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n\n  We next consider the case \\(m^2 < \\infty\\).\n  Suppose first that \\(f\\) is bounded.\n  We write\n  \\begin{align}\n    N_t^{a}\n     & =\n    1_{\\cbra{t<T_a}}\n    \\rbra{h(X_t)+h(-a)-h(X_t-a)}f(L_t)                 \\\\\n     & \\qquad +\n    1_{\\cbra{t<T_a}}\n    \\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_a)\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)}f(L_t+u)\n    \\, \\mathrm{d} u                                           \\\\\n     & \\eqqcolon {(\\mathrm{I})}_a + {(\\mathrm{II})}_a, \\\\\n    M_t^{{(\\pm 1)}}\n     & = h^{(\\pm 1)}(X_t) f(L_t)\n    + \\int_0^\\infty f(L_t+u) \\, \\mathrm{d} u\\label{eq:useless-dot}         \\\\\n     & \\eqqcolon {(\\mathrm{I})} + {(\\mathrm{II})}.\n  \\end{align}\n  Since \\(h\\) is subadditive, we have\n  \\(h(X_t)+h(-a)-h(X_t-a) \\le h(X_t) + h(-X_t)\\).\n  By the proof of Lemma~\\ref{Lem:h-l1-conv},\n  we know \\(\\ensuremath{\\mathbb{P}}_x\\sbra{h(X_t) + h(-X_t)} < \\infty\\)\n  and thus, by the dominated convergence theorem,\n  we have\n  \\begin{align}\n    h(X_t)+h(-a)-h(X_t-a)\n    \\longrightarrow\n    h^{(\\pm 1)}(X_t),\n    \\quad \\text{in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).}\n  \\end{align}\n  The boundedness of \\(f\\) implies that\n  \\({(\\mathrm{I})}_a \\to (\\mathrm{I})\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n  Since \\({(\\mathrm{II})}_a \\le \\int_0^\\infty f(u)\\, \\mathrm{d} u\\),\n  we may apply the dominated convergence theorem to conclude\n  \\({(\\mathrm{II})}_a \\to (\\mathrm{II})\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n  Hence we obtain \\(N_t^{a}\\to M_t^{(\\pm 1)}\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  By Theorem~\\ref{Thm:property-h}\n  and the optional stopping theorem,\n  we obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{A_t^{a}}\n     & =\n    h^B(a) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a}); T_a \\le t}    \\\\\n     & =\n    \\frac{h^B(a)}{a}\\frac{a}{h(a)}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{h(X_{T_a})f(L_{T_a}); T_a \\le t} \\\\\n     & \\le\n    \\frac{h^B(a)}{a}\\frac{a}{h(a)}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_{T_a}^{(0)}; T_a \\le t}        \\\\\n     & =\n    \\frac{h^B(a)}{a}\\frac{a}{h(a)}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(0)}; T_a \\le t}            \\\\\n     & \\to 0,\n    \\qquad \\text{as \\(a\\to\\pm\\infty\\).}\n  \\end{align}\n  This yields that \\(A_t^{a} \\to 0\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n  Hence \\(M_t^{a} \\to  M_t^{(\\pm 1)}\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n  Since \\(\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^a|\\ensuremath{\\mathcal{F}}_s}=M_s^a\\) for \\(0\\le s\\le t\\),\n  we let \\(a\\to\\pm\\infty\\) to obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}|\\ensuremath{\\mathcal{F}}_s}=M_s^{(\\pm 1)}.\\label{eq:M^1-mart}\n  \\end{align}\n  In particular, \\(( M_t^{(\\pm 1)}, t \\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  To remove the boundedness condition of \\(f\\), we consider \\(f\\wedge n\\) and\n  let \\(n\\to\\infty\\) in~\\eqref{eq:M^1-mart}. Then we have\n  \\(( M_t^{(\\pm 1)}, t \\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  The remainder of the proof is the same as that of\n  Theorem~\\ref{Thm:exp-L^1-conv}. So we omit it.\n\\end{proof}\n\n\\begin{Rem}\\label{Rem:martingale}\n  Suppose that \\(m^2 < \\infty\\).\n  Then, since \\(M_t^{(0)}\\) and \\(M_t^{(1)}\\)\n  are different \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingales,\n  \\(\n  m^2 \\rbra{M_t^{(1)} - M_t^{(0)}}\n  = X_t f(L_t)\n  \\)\n  is also a \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  In fact, if \\(X_1\\) is integrable, \\(\\ensuremath{\\mathbb{P}}\\sbra{X_1} = 0\\) and\n  \\(f\\) is a bounded measurable function, then\n  \\((X_t f(L_t),t\\ge 0)\\) is a \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  For more details, see Theorem~\\ref{Thm:mart-XfL}.\n\\end{Rem}\n\nBy Remark~\\ref{Rem:martingale}, we can prove Theorem~\\ref{Thm:martingale}.\n\\begin{proof}[Proof of Theorem~\\ref{Thm:martingale}]\n  Since \\(h^{(\\gamma)}\\) is non-negative for \\(-1\\le \\gamma\\le 1\\),\n  we have \\(M_t^{(\\gamma)}\\ge 0\\).\n  Since \\((M_t^{(0)},t\\ge 0)\\) and \\((X_t f(L_t),t\\ge 0)\\) are\n  \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingales, the process\n  \\((M_t^{(\\gamma)}=M_t^{(0)}+\\frac{\\gamma}{m^2} X_t f(L_t),t\\ge 0)\\)\n  is also a \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{proof}\n\n\\section{Local time penalization with two-point hitting time\n    clock}\\label{Sec:two-hitting-time}\nLet us consider the hitting time of\ntwo-point set, i.e.,\n\\(T_{a,b}=T_{\\cbra{a, b}} = T_a \\wedge T_b\\).\nRecall that we assume \\(X\\) is recurrent\nand assume the condition~\\ref{item:assumption}.\n\\subsection{The law of the local time with two-point hitting time clock}\nFirst, we compute \\(\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a \\wedge T_b}},  \\ensuremath{\\mathbb{P}}_x(T_a<T_b\\wedge T_c)\\)\nand \\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a\\wedge T_b})}\\) respectively.\n\\begin{Lem}\\label{Lem:LT-two-hitting}\n  For \\(a\\ne b\\), it holds that\n  \\begin{align}\n    h^C(a,b)\n     & \\coloneqq\n    \\ensuremath{\\mathbb{P}}\\sbra{L_{T_a \\wedge T_b}} \\\\\n     & =\\frac{1}{h^B(a-b)}\n    \\cbra*{\\begin{multlined}\n        \\rbra[\\big]{h(b)+h(-a)}h(a-b)\n        +\\rbra[\\big]{h(a)+h(-b)}h(b-a)\\\\\n        -h(a-b)h(b-a)\n      \\end{multlined}}.\n  \\end{align}\n\\end{Lem}\n\\begin{proof}\n  For \\(q>0\\), by the strong Markov property, we have\n  \\begin{align}\\label{eq:L_T_aT_b}\n    \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt}\\, \\mathrm{d} L_t}\n    =\\begin{aligned}[t]\n       & \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^{T_a\\wedge T_b}\\mathrm{e}^{-qt}\\, \\mathrm{d} L_t }\n      \\\\\n       & + \\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}\\ensuremath{\\mathbb{P}}_a\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt}\\,\\mathrm{d} L_t\n      }\n      \\\\\n       & + \\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{-qT_b}; T_b<T_a}\\ensuremath{\\mathbb{P}}_b\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt}\\,\\mathrm{d} L_t\n      }.\n    \\end{aligned}\n  \\end{align}\n  Using~\\eqref{eq:regularity-of-L} and\n  (\\ref{Lem-item:two-hittig-time-prob}) of Lemma~\\ref{Lem:h^B-hitting-h}\n  and letting \\(q\\to 0+\\),\n  we obtain the desired result.\n\\end{proof}\n\n\\begin{Lem}\\label{Lem:three-hitting-time}\n  For \\(a,b,c,x\\in \\ensuremath{\\mathbb{R}}\\) and \\(a\\ne b, c\\), it holds that\n  \\begin{align}\\label{eq:e-three-hitting-time}\n   \\hspace{-25pt} \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b\\wedge T_c}\n    = \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}-\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_c}; T_c< T_b}\n      -\\ensuremath{\\mathbb{P}}_c\\sbra{\\mathrm{e}^{-qT_a}; T_a< T_b} }\n    {1-\\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-qT_c};T_c<T_b}\\ensuremath{\\mathbb{P}}_c\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}},\n  \\end{align}\n  and letting \\(q\\to 0+\\), it holds that\n  \\begin{align}\\label{eq:three-hitting-time}\n    \\ensuremath{\\mathbb{P}}_x(T_a<T_b\\wedge T_c)\n    = \\frac{\\ensuremath{\\mathbb{P}}_x(T_a<T_b)-\\ensuremath{\\mathbb{P}}_x(T_c<T_b)\\ensuremath{\\mathbb{P}}_c(T_a<T_b)}\n    {1-\\ensuremath{\\mathbb{P}}_a(T_c<T_b)\\ensuremath{\\mathbb{P}}_c(T_a<T_b)}.\n  \\end{align}\n\\end{Lem}\nNote that, by~\\eqref{eq:two-hittig-time-prob},\nwe can express\n\\(\\ensuremath{\\mathbb{P}}_x(T_a<T_b\\wedge T_c)\\) only in terms of \\(h\\).\n\\begin{proof}[Proof of Lemma~\\ref{Lem:three-hitting-time}]\n  Using the strong Markov property, we have\n  \\begin{align}\\label{eq:pf-three-hitting-times}\n    \\begin{aligned}\n     & \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}    \\\\\n     & =\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a};T_a<T_b\\wedge T_c}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_c<T_a<T_b} \\\\\n     & =\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a};T_a<T_b\\wedge T_c}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_c}; T_c<T_a\\wedge T_b}\n    \\ensuremath{\\mathbb{P}}_c\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}.\n    \\end{aligned}\n  \\end{align}\n  Replacing \\(a\\) with \\(c\\), we also have\n  \\begin{align}\\label{eq:pf-three-hitting-times2}\n    \\begin{aligned}\n       & \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_c}; T_c<T_b} \\\\\n       & =\n      \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_c};T_c<T_a\\wedge T_b}\n      + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b\\wedge T_c}\n      \\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-qT_c}; T_c<T_b}.\n    \\end{aligned}\n  \\end{align}\n  Combining~\\eqref{eq:pf-three-hitting-times} and~\\eqref{eq:pf-three-hitting-times2},\n  we obtain~\\eqref{eq:e-three-hitting-time}.\n  Letting \\(q\\to 0+\\), we also have~\\eqref{eq:three-hitting-time}.\n\\end{proof}\n\n\\begin{Lem}\\label{Lem:two-hitting-time-f-L}\n  For \\(a \\ne b\\) and \\(x\\in \\ensuremath{\\mathbb{R}}\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a\\wedge T_b})}\n    =\n    \\ensuremath{\\mathbb{P}}_x(T_0 > T_a\\wedge T_b) f(0)\n    + \\frac{\\ensuremath{\\mathbb{P}}_x(T_0 < T_a\\wedge T_b)}{h^C(a,b)}\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^C(a,b)}f(u)\n    \\, \\mathrm{d} u.\n  \\end{align}\n\\end{Lem}\n\\begin{proof}\n  In the same way as~\\eqref{eq:L_T_a-exp-distrib},\n  we have, for \\(l>0\\),\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}(L_{T_a\\wedge T_b}>l)=\n    \\ensuremath{\\mathbb{P}}(\\sigma_{\\cbra{T_a\\wedge T_b <T_0}}>l)\n    = \\mathrm{e}^{-tn(T_a\\wedge T_b <T_0)}.\n  \\end{align}\n  In particular,\n  \\(L_{T_a\\wedge T_b}\\) is exponentially distributed.\n  (Consequently, we obtain \\(n(T_a\\wedge T_b<T_0)=1\/\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a,b}}=1\/h^C(a,b)\\).)\n  We may apply the strong Markov property and obtain the desired result.\n\\end{proof}\n\n\\subsection{Proof of\n  Theorem~\\ref{Thm:two-point-hitting-time-result}}\\label{Subsec:pf-two-hitting}\nWe are now ready to prove the two-point hitting time result.\nRecall that \\((a,b)\\xrightarrow[]{\\gamma}\\infty\\) means~\\eqref{eq:a-b-inf-notation}.\n\n\\begin{proof}[Proof of Theorem~\\ref{Thm:two-point-hitting-time-result}]\n  By Lemmas~\\ref{Lem:LT-two-hitting} and~\\ref{Lem:three-hitting-time}\n  and~\\eqref{eq:two-hittig-time-prob}, we have\n  \\begin{align}\n     & h^C(a,b)\\ensuremath{\\mathbb{P}}_x(T_0>T_a\\wedge T_b)\n    \\\\\n     & =h(x) + \\frac{1}{h^B(a-b)}\n    \\cbra*{\\begin{multlined}\n        \\rbra[\\big]{h(-a)-h(x-a)}h(a-b)\n        + \\rbra[\\big]{h(-b)-h(x-b)}h(b-a)\\\\\n        -\\rbra[\\big]{h(a)-h(b)}\\rbra[\\big]{h(-a)-h(x-a)-h(-b)+h(x-b)}\n      \\end{multlined}}.\n  \\end{align}\n  Recall that \\(h^B(a)=h(a)+h(-a)\\); see~\\eqref{eq:h^B}.\n  Replacing \\(b\\) with \\(-b\\) and\n  using Theorem~\\ref{Thm:property-h}, it holds that\n  \\begin{align}\n    h^C(a, -b) \\ensuremath{\\mathbb{P}}_x(T_0>T_{a}\\wedge T_{-b})\n    \\xrightarrow[(a,b)\\xrightarrow{\\gamma}\\infty]{}\n    h^{(\\gamma)}(x).\n  \\end{align}\n  By the strong Markov property and Lemma~\\ref{Lem:two-hitting-time-f-L},\n  we have\n  \\begin{align}\n    N_t^{a,b} & = 1_{\\cbra{t<T_{a,-b}}}\n    \\cbra*{h^C(a,-b)\\ensuremath{\\mathbb{P}}_{X_t}(T_0>T_{a,-b})f(L_t)} \\\\\n              & \\quad +1_{\\cbra{t<T_{a,-b}}}\n    \\cbra*{\\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_{a,-b})\\int_0^\\infty \\mathrm{e}^{-u\/h^C(a,-b)} f(L_t+u)\\, \\mathrm{d} y}.\n  \\end{align}\n  Hence we obtain\n  \\(N_t^{a,b}\\to M_t^{(\\gamma)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\((a,b)\\xrightarrow[]{\\gamma}\\infty\\).\n  The proof of \\(M_t^{a,b}\\to M_t^{(\\gamma)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\\n  is similar to~\\eqref{eq:hitting-A}--\\eqref{eq:hitting-Al}.\n  Since \\((M_t^{(\\gamma)},t\\ge 0)\\) is\n  a non-negative \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale (Theorem~\\ref{Thm:martingale}),\n  we may apply Scheff\\'{e}'s lemma to obtain\n  the \\(\\ensuremath{\\mathcal{L}}^1\\) convergence.\n\\end{proof}\n\n\\section{Local time penalization with inverse local time clock}\\label{Sec:inv-time}\nWe define the modified Bessel function of the first kind, which\nis expressed as\n\\begin{align}\n  I_{\\nu}(x) = \\sum_{n=0}^\\infty \\frac{{(x\/2)}^{\\nu+2n}}{n! \\Gamma(\\nu+n+1)},\n  \\quad \\nu \\ge 0, \\,\\, x > 0.\n\\end{align}\nNote that \\(I_{\\nu}(x)\\) is increasing in \\(x>0\\).\nFor more details, see e.g.,~\\cite[Section 5]{MR0350075}.\nRecall that \\(\\eta_u^a\\) denotes the inverse local time at \\(a\\):\n\\(\n\\eta_u^a = \\inf\\cbra{t\\ge 0\\colon L_t^a > u}\n\\).\nWe consider the penalization with inverse local time clock\nin two ways:\nfirst, we make\n\\(a\\) tend to infinity, and second, \\(u\\) tend to infinity.\nRecall that we assume \\(X\\) is recurrent\nand assume the condition~\\ref{item:assumption}.\n\n\\subsection{The law of the local time with inverse local time clock}\n\\begin{Lem}\n  Let \\(a \\in \\ensuremath{\\mathbb{R}}\\setminus\\cbra{0}\\).\n  Then the process \\((L_{\\eta^a_u},u\\ge 0)\\) under \\(\\ensuremath{\\mathbb{P}}_a\\)\n  is a compound Poisson process with Laplace transform\n  \\begin{align}\\label{eq:L_eta^a-laplace}\n    \\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-\\beta L_{\\eta^a_u}}}\n    = \\mathrm{e}^{-u\\beta\/(1+\\beta h^B(a))}, \\quad \\beta\\ge 0.\n  \\end{align}\n  Moreover, for any \\(u>0\\) and \\(f\\in\\ensuremath{\\mathcal{L}}^1_{+}\\),\n  it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_a\\sbra{f(L_{\\eta^a_u})}\n    = \\mathrm{e}^{-u\/h^B(a)} f(0)\n    + \\int_0^\\infty f(y)\\rho^{h^B(a)}_u(y)\n    \\, \\mathrm{d} y,\n  \\end{align}\n  where\n  \\begin{align}\n    \\rho^a_u(y) = \\mathrm{e}^{-(u+y)\/a}\\frac{\\sqrt{u\/y}}{a}\n    I_1\\rbra*{\\frac{2\\sqrt{uy}}{a}}.\n  \\end{align}\n\\end{Lem}\nWe omit the proof because it is very similar to that in the\ndiffusion case of Profeta--Yano--Yano~\\cite[Lemma 4.1]{MR3909919}, using\nTheorem~\\ref{Thm:n-two-hitting-time}.\nFor the proof, we use\n\\(n^a(T_0<T_a) = n(T_{-a}<T_0) = 1\/h^B(a)\\)\nin the L\\'{e}vy case, instead of\n\\(n^a(T_0<T_a) = n(T_a<T_0) = 1\/a\\)\nin the diffusion case.\nThe proof of~\\eqref{eq:L_eta^a-laplace} can also be found\nin~\\cite[Lemma V.13]{MR1406564}.\n\nThe proof of the next lemma is completely parallel\nto that of~\\cite[Lemma 4.2]{MR3909919}.\nSo we omit it.\n\\begin{Lem}\n  For \\(u > 0, \\; x,a \\in\\ensuremath{\\mathbb{R}}\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\eta^a_u})}\n     & = \\ensuremath{\\mathbb{P}}_x(T_a<T_0)\\ensuremath{\\mathbb{P}}_a\\sbra{f(L_{\\eta^a_u})}\n    + \\ensuremath{\\mathbb{P}}_x(T_0<T_a)\\ensuremath{\\mathbb{P}}_a\\sbra{f(\\bm{e}_{1\/h^B(a)}+L_{\\eta^a_u})} \\\\\n     & = \\ensuremath{\\mathbb{P}}_x(T_a<T_0)\\ensuremath{\\mathbb{P}}_a\\sbra{f(L_{\\eta^a_u})}\n    + \\frac{\\ensuremath{\\mathbb{P}}_x(T_0<T_a)}{h^B(a)}\n    \\int_0^\\infty f(y)\\widetilde{\\rho}^{h^B(a)}_u(y)\n    \\, \\mathrm{d} y,\n  \\end{align}\n  where\n  \\begin{align}\n    \\widetilde{\\rho}^a_u(y)\n    = \\mathrm{e}^{-(u+y)\/a}I_0\\rbra*{\\frac{2\\sqrt{uy}}{a}}.\n  \\end{align}\n\\end{Lem}\n\n\\subsection{Limit as \\(a\\) tends to infinity with \\(u\\) being fixed}\n\\begin{Thm}\\label{Thm:inv-LT-result}\n  Let \\(f\\in \\ensuremath{\\mathcal{L}}^1_{+}\\) and \\(x\\in\\ensuremath{\\mathbb{R}}\\).\n  For any \\(u> 0\\) and \\(a\\in \\ensuremath{\\mathbb{R}}\\), we define\n  \\begin{align}\n    N_t^{a,u}\n     & = h^B(a)\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\eta^a_u}); t<\\eta^a_u|\\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^{a,u}\n     & = h^B(a)\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\eta^a_u})|\\ensuremath{\\mathcal{F}}_t}.\n  \\end{align}\n  Then\n  \\begin{align}\n    \\lim_{a\\to\\pm\\infty}N_t^{a,u}\n    = \\lim_{a\\to\\pm\\infty}M_t^{a,u}\n    = M_t^{(\\pm 1)},\\quad\n    \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n  Consequently, if \\(M_0^{(\\pm 1)}>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\eta^a_u})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\eta^a_u})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t^{(\\pm 1)}}{M_0^{(\\pm 1)}}},\n    \\qquad \\text{as \\(a \\to \\pm\\infty \\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\nThe proof of the theorem is\nvery similar to that of~\\cite[Lemma 4.4 and Theorem 4.5]{MR3909919}.\nSo we omit it.\n(\\cite[Lemma 4.4]{MR3909919} states only convergence\nin probability but, its a.s.\\ convergence\ncan also be proved by the same proof.)\n\n\\subsection{Limit as \\(u\\) tends to infinity with \\(a\\) being fixed}\nIn this section, we only consider the cases \\(f(x) = \\mathrm{e}^{-\\beta x}\\)\nand \\(f(x) = 1_{\\cbra{x=0}}\\).\nThe next theorem is in the case \\(f(x)= \\mathrm{e}^{-\\beta x}\\).\n\\begin{Thm}\\label{Thm:inv-u-e}\n  Let \\(x \\in \\ensuremath{\\mathbb{R}}\\), \\(a \\in \\ensuremath{\\mathbb{R}}\\setminus\\cbra{0}\\), \\(\\beta>0\\) and \\(t>0\\).\n  Define\n  \\begin{align}\n    N_t^{u,\\beta, a} & =\n    \\mathrm{e}^{\\beta u\/(1+\\beta h^B(a))}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-\\beta L_{\\eta^a_u}}; t<\\eta^a_u | \\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^{u,\\beta, a} & =\n    \\mathrm{e}^{\\beta u\/(1+\\beta h^B(a))}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-\\beta L_{\\eta^a_u}} | \\ensuremath{\\mathcal{F}}_t},\n  \\end{align}\n  and\n  \\begin{align}\n    M_t^{\\beta,a}\n    = \\mathrm{e}^{-\\beta L_t}\n    \\cbra*{\\ensuremath{\\mathbb{P}}_{X_t}(T_a<T_0)\n      + \\frac{\\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_a)}{1+\\beta h^B(a)}\\mathrm{e}^{\\beta L_t^a\/(1+\\beta h^B(a))}}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\lim_{u\\to\\infty}N_t^{u,\\beta,a}\n    =\\lim_{u\\to\\infty}M_t^{u,\\beta,a}\n    = M_t^{\\beta,a},\\quad\n    \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\\n      and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:inv-u-e} is parallel\nto that of~\\cite[Lemma 4.6 and Theorem 4.7]{MR3909919}.\nSo we omit it.\n\nWe consider the case \\(f(x) = 1_{\\cbra{x=0}}\\).\n\\begin{Thm}\\label{Thm:inv-u-1}\n  Let \\(x \\in \\ensuremath{\\mathbb{R}}\\), \\(a \\in \\ensuremath{\\mathbb{R}}\\setminus\\cbra{0}\\), \\(\\beta>0\\) and \\(t>0\\).\n  Define\n  \\begin{align}\n    N_t^{u,\\infty,a}\n     & = \\mathrm{e}^{u\/h^B(a)}\n    \\ensuremath{\\mathbb{P}}_x\\rbra{t<\\eta_u^a<T_0|\\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^{u,\\infty,a}\n     & = \\mathrm{e}^{u\/h^B(a)}\n    \\ensuremath{\\mathbb{P}}_x\\rbra{\\eta_u^a<T_0|\\ensuremath{\\mathcal{F}}_t},\n  \\end{align}\n  and\n  \\begin{align}\n    M_t^{\\infty,a}\n    = \\mathrm{e}^{L_t^a\/h^B(a)} \\ensuremath{\\mathbb{P}}_{X_t}(T_a<T_0) 1_{\\cbra{t<T_0}}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\lim_{u\\to\\infty}N_t^{u,\\infty,a}\n    =\\lim_{u\\to\\infty}M_t^{u,\\infty,a}\n    =M_t^{\\infty,a},\\quad\n    \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\\n      and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:inv-u-1} is very similar to\nthat of~\\cite[Theorem 4.8]{MR3909919}.\nSo we omit it.\n\n\\section{Universal \\(\\sigma\\)-finite measures}\\label{Sec:universal-measure}\nIn this section, we shall discuss the penalized processes and\n\\(\\sigma\\)-finite measures unifying the processes.\nWe have obtained the penalized measure \\(\\ensuremath{\\mathbb{Q}}^{(\\gamma,f)}_x\\),\nwhich is given by\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}^{(\\gamma,f)}_x|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\frac{M_t^{(\\gamma,f)}}{M_0^{(\\gamma,f)}}\\cdot\\ensuremath{\\mathbb{P}}_x|_{\\ensuremath{\\mathcal{F}}_t},\n  \\quad -1\\le \\gamma \\le 1.\n\\end{align}\n\n\\subsection{L\\'{e}vy processes conditioned to avoid zero}\nWe show that \\(h^{(\\gamma)}\\) is invariant for the killed process.\n\\begin{Thm}\\label{Thm:hg-invariant}\n  For \\(-1\\le \\gamma\\le 1\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{h^{(\\gamma)}(X_t);T_0>t} = h^{(\\gamma)}(x), \\quad\n    n\\sbra{h^{(\\gamma)}(X_t);T_0>t} = 1, \\quad\n    x \\in \\ensuremath{\\mathbb{R}}.\n  \\end{align}\n\\end{Thm}\n\\begin{proof}\n  We can show the case \\(\\gamma=0\\)\n  by the completely same discussion as the proof\n  of Pant\\'{\\i}~\\cite[(iii) of Theorem 2.2]{MR3689384}.\n  Combining this with Theorem~\\ref{Thm:mart-XfL}, we obtain the desired result.\n  The former equation also follows from the fact that\n  \\((M_t^{(\\gamma, 1_{\\cbra{u=0}})},t\\ge 0)\\) is a \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{proof}\n\nLet \\(\\ensuremath{\\mathcal{H}}^{(\\gamma)} = \\cbra{x\\in\\ensuremath{\\mathbb{R}}\\colon h^{(\\gamma)}(x)>0 }\\) and\n\\(\\ensuremath{\\mathcal{H}}^{(\\gamma)}_0 = \\ensuremath{\\mathcal{H}}^{(\\gamma)} \\cup \\cbra{0}\\).\nWe introduce the \\(h^{(\\gamma)}\\) transformed process given by\n\\begin{align}\\label{eq:h-g-trans}\n  \\ensuremath{\\mathbb{P}}_x^{(\\gamma)}|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\begin{dcases}\n    1_{\\cbra{T_0>t}} \\frac{h^{(\\gamma)}(X_t)}{h^{(\\gamma)}(x)}\n    \\cdot \\ensuremath{\\mathbb{P}}_x|_{\\ensuremath{\\mathcal{F}}_t}               & \\text{if } x\\in\\ensuremath{\\mathcal{H}}^{(\\gamma)}, \\\\\n    1_{\\cbra{T_0>t}}  h^{(\\gamma)}(X_t) \\cdot n|_{\\ensuremath{\\mathcal{F}}_t} & \\text{if } x = 0.\n  \\end{dcases}\n\\end{align}\nSince \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}|_{\\ensuremath{\\mathcal{F}}_t}\\) is consistent in \\(t>0\\),\nthe probability measure \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\) can be well-defined\non \\(\\ensuremath{\\mathcal{F}}_\\infty\\coloneqq \\sigma(X_t,t\\ge 0)\\),\nfor more details, see Yano~\\cite[Theorem 9.1]{yano2021universality}.\nFor any \\(t>0\\), we have\n\\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\rbra{T_{\\ensuremath{\\mathbb{R}} \\setminus \\ensuremath{\\mathcal{H}}^{(\\gamma)} } > t} = 1\\).\nConsequently, we have\n\\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\rbra{T_{\\ensuremath{\\mathbb{R}} \\setminus\\ensuremath{\\mathcal{H}}^{(\\gamma)}} = \\infty} = 1\\) and\nin particular, \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\rbra{T_0= \\infty}=1\\).\nThe process \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\)\nis called a \\textit{L\\'{e}vy process conditioned to avoid zero}.\nNote that, for \\(x\\in\\ensuremath{\\mathcal{H}}^{(\\gamma)}\\),\nthe measure \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\) is absolutely continuous with respect to\n\\(\\ensuremath{\\mathbb{P}}_x\\) on \\(\\ensuremath{\\mathcal{F}}_t\\), but is\nsingular to \\(\\ensuremath{\\mathbb{P}}_x\\) on \\(\\ensuremath{\\mathcal{F}}_\\infty\\) since \\(\\ensuremath{\\mathbb{P}}_x(T_0<\\infty)=1\\).\n\nBy Theorems~\\ref{Thm:exp-time-result},~\\ref{Thm:hitting-time-result},~\\ref{Thm:two-point-hitting-time-result}\nand~\\ref{Thm:inv-LT-result}\n(Corollaries~\\ref{Cor:hitting-cond} and~\\ref{Cor:two-point-hitting-cond})\nand by taking \\(f=1_{\\cbra{u=0}}\\),\nwe have the following conditioning results.\n\\begin{Cor}\\label{Cor:avoid-zero}\n  Let \\(t>0\\) and \\(F_t\\) be a bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functional.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item\n    \\(\\displaystyle\n    \\lim_{q\\to 0+}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>\\bm{e}_q}=\\ensuremath{\\mathbb{P}}_x^{(0)}\\sbra{F_t}\n    \\), for \\(x\\in \\ensuremath{\\mathcal{H}}^{(0)}\\);\\label{Cor-item:avoid-zero-exp}\n    \\item\n    \\(\\displaystyle\n      \\lim_{a\\to\\pm\\infty}\n      \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>T_a}=\\ensuremath{\\mathbb{P}}_x^{(\\pm 1)}\\sbra{F_t}\n    \\), for \\(x\\in \\ensuremath{\\mathcal{H}}^{(\\pm 1)}\\);\n    \\item\n    \\(\\displaystyle\n      \\lim_{(a,b)\\xrightarrow[]{\\gamma}\\infty}\n      \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>T_{a,-b}}=\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\sbra{F_t}\n    \\), for \\(-1\\le \\gamma\\le 1\\) and \\(x\\in \\ensuremath{\\mathcal{H}}^{(\\gamma)}\\);\n    \\item\n    \\(\\displaystyle\n      \\lim_{a\\to\\pm\\infty}\n      \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>\\eta^a_u}=\\ensuremath{\\mathbb{P}}_x^{(\\pm 1)}\\sbra{F_t}\n    \\), for \\(u>0\\) and \\(x\\in \\ensuremath{\\mathcal{H}}^{(\\pm 1)}\\).\n  \\end{enumerate}\n\\end{Cor}\nNote that (\\ref{Cor-item:avoid-zero-exp}) of Corollary~\\ref{Cor:avoid-zero}\ngeneralizes Pant\\'{\\i}~\\cite[Theorem 2.7]{MR3689384}.\n\n\\subsection{Universal \\(\\sigma\\)-finite measures}\nIn this subsection, we assume that \\((X,\\ensuremath{\\mathbb{P}}_x)\\) has\na transition density \\(p_t(\\cdot)\\).\nThen we can construct the L\\'{e}vy bridge.\nLet \\(\\ensuremath{\\mathbb{P}}_{x,y}^u\\) denote the law of bridge\nfrom \\(X_0=x\\) to \\(X_u=y\\).\nThis measure can be constructed as\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_{x,y}^u(A) = \\ensuremath{\\mathbb{P}}_x\\sbra*{1_A\\frac{p_{u-t}(y-X_t)}{p_t(y-x)}},\n  \\quad A\\in\\ensuremath{\\mathcal{F}}_t,\\; 0< t<u.\n\\end{align}\nSee Fitzsimmons--Pitman--Yor~\\cite{MR1278079}.\nWe have the conditioning formula:\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^t F_u \\, \\mathrm{d} L_u}\n  = \\int_0^t \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{d} L_u}\\ensuremath{\\mathbb{P}}_{x, 0}^{u}\\sbra{F_u},\n  \\quad t>0,\n\\end{align}\nfor all non-negative predictable processes \\((F_u)\\), where\nwe write symbolically\n\\( \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{d} L_u} = p_u(-x) \\, \\mathrm{d} u\\).\n\nFor \\(x\\in \\ensuremath{\\mathbb{R}}\\) and \\(-1\\le \\gamma\\le 1\\), we define\n\\begin{align}\n  \\ensuremath{\\mathcal{P}}_x^{(\\gamma)}\n   & = \\int_0^\\infty \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{d} L_u}\n  \\rbra*{\\ensuremath{\\mathbb{P}}_{x,0}^{u} \\bullet \\ensuremath{\\mathbb{P}}_0^{(\\gamma)}}\n  + h^{(\\gamma)}(x)\\ensuremath{\\mathbb{P}}_x^{(\\gamma)},\n\\end{align}\nwhere\nthe symbol \\(\\bullet\\) stands for the concatenation and\n\\(h^{(\\gamma)}(x)\\ensuremath{\\mathbb{P}}_x^{(\\gamma)} = 0\\)\nfor \\(x \\in \\ensuremath{\\mathbb{R}}\\setminus\\ensuremath{\\mathcal{H}}^{(\\gamma)}\\).\nThen we have the following:\n\\begin{Thm}\n  Let \\(x \\in \\ensuremath{\\mathbb{R}}\\) and \\(f\\in \\ensuremath{\\mathcal{L}}^1_{+}\\).\n  Let \\(t>0\\) and \\(F_t\\) be a bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable\n  functional.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item\n          \\(\\displaystyle\n          \\lim_{q\\to 0+}\n          r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\bm{e}_q})}\n          = \\ensuremath{\\mathcal{P}}_x^{(0)} \\sbra{F_t f(L_{\\infty})}\\);\n    \\item\n          \\(\\displaystyle\n          \\lim_{a\\to\\pm\\infty}\n          h^B(a)\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{T_a})}\n          = \\ensuremath{\\mathcal{P}}_x^{(\\pm 1)} \\sbra{F_t f(L_{\\infty})}\\);\n    \\item\n          \\(\\displaystyle\n          \\lim_{(a,b)\\xrightarrow[]{\\gamma}\\infty}\n          h^C(a, -b)\n          \\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{T_{a, -b}})}\n          =\\ensuremath{\\mathcal{P}}_x^{(\\gamma)} \\sbra{F_t f(L_{\\infty})}\\),\n          for \\(-1\\le\\gamma\\le 1\\);\n    \\item\n          \\(\\displaystyle\n          \\lim_{a\\to\\pm\\infty}\n          h^B(a)\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\eta^a_u})}\n          = \\ensuremath{\\mathcal{P}}_x^{(\\pm 1)} \\sbra{F_t f(L_{\\infty})}\\), for \\(u>0\\).\n  \\end{enumerate}\n\\end{Thm}\n\\begin{proof}\n  It suffices to show that\n  \\begin{align}\n    \\ensuremath{\\mathcal{P}}^{(\\gamma)}_x\\sbra{F_t f(L_\\infty)} = \\ensuremath{\\mathbb{P}}_x\\sbra{F_t M_t^{(\\gamma)}},\n  \\end{align}\n  for \\(-1 \\le \\gamma \\le 1\\).\n  The proof is the same as that of Theorem 5.3 of~\\cite{MR3909919}.\n\\end{proof}\nConsequently, we obtain the representation of \\(\\ensuremath{\\mathbb{Q}}^{(\\gamma,f)}_x\\) as follows:\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}^{(\\gamma,f)}_x\n  = \\frac{f(L_\\infty)}{\\ensuremath{\\mathcal{P}}^{(\\gamma)}_x\\sbra{f(L_\\infty)}}\n  \\cdot \\ensuremath{\\mathcal{P}}^{(\\gamma)}_x.\n\\end{align}\nRecall that \\(g = \\sup\\{t\\colon X_t=0\\}\\).\nSince \\(\\ensuremath{\\mathbb{O}}^{(\\gamma,f)}_x(g < \\infty) = \\ensuremath{\\mathbb{P}}_x(g = \\infty)= 1\\),\nthe two measures are singular on \\(\\ensuremath{\\mathcal{F}}_\\infty\\).\n\n\\subsection{The law of \\(L_\\infty\\) under \\(\\ensuremath{\\mathbb{Q}}_0^{(\\gamma,f)}\\)}\nWe assume,\nfor simplicity, that \\(f\\in \\ensuremath{\\mathcal{L}}^1_{+}\\)\nsatisfies \\(\\int_0^\\infty f(u)\\, \\mathrm{d} u = 1\\).\nThen we have \\(M_0^{(\\gamma,f)}=1\\), \\(\\ensuremath{\\mathbb{P}}_0\\)-a.s.\nFor \\(l\\ge 0\\), the optional stopping theorem implies that\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}_0^{(\\gamma,f)}(L_t\\ge l)\n  = \\ensuremath{\\mathbb{P}}_0\\sbra{M_t^{(\\gamma, f)}; L_t\\ge l}\n  = \\ensuremath{\\mathbb{P}}_0\\sbra{M_{\\eta_l}^{(\\gamma, f)}; \\eta_l\\le t}.\n\\end{align}\nLetting \\(t\\to\\infty\\), we may apply the monotone convergence theorem to\ndeduce that\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}_0^{(\\gamma,f)}(L_\\infty\\ge l) = \\ensuremath{\\mathbb{P}}_0\\sbra{M_{\\eta_l}^{(\\gamma, f)}}.\n\\end{align}\nSince \\(X_{\\eta_l}= 0\\) and \\(L_{\\eta_l}=l\\), we have\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_0\\sbra{M_{\\eta_l}^{(\\gamma, f)}}\n  = \\int_0^\\infty f(l+u)\\, du.\n\\end{align}\nTherefore, it holds that\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}_0^{(\\gamma,f)}(L_\\infty \\in \\mathrm{d} u) = f(u)\\, \\mathrm{d} u,\n  \\quad u>0.\n\\end{align}\n\n\\section{The transient case}\\label{Sec:trans}\nWe now study penalization in the transient case.\nThroughout this section, we always assume\n\\(X\\) is transient and\nthe conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\nRecall that\n\\begin{align}\n  \\kappa\\coloneqq \\lim_{q\\to 0+}\\frac{1}{r_q(0)}\n  = n(T_0=\\infty)>0;\n\\end{align}\nsee~\\eqref{eq:kappa}.\n\\subsection{The renormalized zero resolvent in the transient case}\nAs in the recurrent case, we define \\(h_q(x)=r_q(0)-r_q(-x)\\).\n\\begin{Thm}\n  Suppose that\n  the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\n  Then the following assertions hold.\n  \\begin{enumerate}\n    \\item For any \\(x\\in\\ensuremath{\\mathbb{R}}\\), it holds that\n          \\(\\displaystyle\n          h(x)\\coloneqq \\lim_{q\\to 0+} h_q(x) = \\kappa^{-1}\\ensuremath{\\mathbb{P}}_x(T_0=\\infty).\n          \\)\n    \\item The above convergence is uniform on compacts,\n          and consequently \\(h\\) is continuous.\\label{Thm-item:h-unif-conv-trans}\n    \\item \\(h\\) is subadditive on \\(\\ensuremath{\\mathbb{R}}\\),\n          that is, \\(h(x+y)\\le h(x)+h(y)\\) for\n          \\(x,y\\in\\ensuremath{\\mathbb{R}}\\).\\label{Thm-item:h-subadd-trans}\n  \\end{enumerate}\n\\end{Thm}\n\n\\begin{proof}\n  It follows from~\\eqref{eq:exp-hit-time}\n  and~\\eqref{eq:kappa} that\n  \\begin{align}\\label{eq:transient-h-conv}\n    h_q(x) = r_q(0)\\ensuremath{\\mathbb{P}}_x\\sbra{1-\\mathrm{e}^{-qT_0}}\n    \\longrightarrow \\kappa^{-1} \\ensuremath{\\mathbb{P}}_x(T_0 = \\infty),\n    \\quad \\text{as \\(q \\to 0+\\).}\n  \\end{align}\n  The proof of (\\ref{Thm-item:h-unif-conv-trans}) and (\\ref{Thm-item:h-subadd-trans})\n  are the same as that of Theorem~\\ref{Thm:exist-h}.\n\\end{proof}\n\n\\begin{Thm}\n  Suppose that\n  the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item\n          \\(\\displaystyle\n          \\lim_{x\\to\\pm\\infty} \\frac{h(x)}{\\abs{x}} =0\n          \\);\\label{Thm-item:h\/x-infinity-trans}\n    \\item\n          \\(\\displaystyle\n          \\lim_{y\\to\\pm\\infty} \\cbra*{h(x+y) - h(y)} =0,\n          \\) for all \\(x\\in\\ensuremath{\\mathbb{R}}\\).\\label{Thm-item:h-h-limit-trans}\n  \\end{enumerate}\n\\end{Thm}\n\\begin{proof}\n  Since \\(h(x)\\le \\kappa^{-1}\\), it is\n  obvious that~\\eqref{Thm-item:h\/x-infinity-trans} holds.\n  The proof of~\\eqref{Thm-item:h-h-limit-trans} is the same as that of\n  (\\ref{Thm-item:h-h-limit}) of Theorem~\\ref{Thm:property-h}\n  in the recurrent and \\(m^2=\\infty\\) case.\n\\end{proof}\n\\subsection{Useful equations}\nBefore stating out penalization result, we introduce\nsome useful equations.\n\\begin{Lem}\\label{Lem:useful-eq-trans}\n  \\begin{enumerate}\n    \\item For \\(a\\in\\ensuremath{\\mathbb{R}}\\),\n          \\begin{align}\n            h^B(a) \\coloneqq \\lim_{q\\to 0+} h_q^B(a)\n            =\\ensuremath{\\mathbb{P}}_0\\sbra{L_{T_a}}= h(a) + h(-a) - \\kappa h(a)h(-a).\n            \\label{eq:h^B-trans}\n          \\end{align}\\label{Lem-item:hq-two-hitting-time-trans}\n    \\item For \\(x,a,b\\in\\ensuremath{\\mathbb{R}}\\) and \\(a\\ne b\\),\n          \\begin{align}\n            \\ensuremath{\\mathbb{P}}_x(T_a < T_b)\n            = \\frac{h(b-a)+h(x-b)-h(x-a) - \\kappa h(x-b)h(b-a)}{h^B(a-b)}.\n            \\label{eq:two-hittig-time-prob-trans}\n          \\end{align}\n    \\item\n          For \\(x,a,b\\in\\ensuremath{\\mathbb{R}}\\) and \\(a\\ne b\\),\n          \\begin{align}\\label{eq:L_T_aT_b-trans}\n            \\ensuremath{\\mathbb{P}}\\sbra{L_{T_a \\wedge T_b}}\n             & =\\frac{1}{h^B(a-b)}\n            \\cbra*{\\begin{multlined}\n                \\rbra[\\big]{h(b)+h(-a)-\\kappa h(-a)h(b)}h(a-b)\t\\\\\n                \\qquad +\\rbra[\\big]{h(a)+h(-b)-\\kappa h(-b)h(a)}h(b-a) \\\\\n                -h(a-b)h(b-a)\n              \\end{multlined}}.\n          \\end{align}\n  \\end{enumerate}\n\\end{Lem}\nThe proof of Lemma~\\ref{Lem:useful-eq-trans} is similar to\nthat in the recurrent case.\nSo we omit it.\n\n\\begin{Lem}[{\\cite[Lemma 3.10]{MR3689384}}]\\label{Thm:h^B-infty-trans}\n  It holds that \\(\\lim_{x\\to\\infty} h^B(x) = \\kappa^{-1}\\).\n\\end{Lem}\n\\begin{proof}\n  For completeness of the paper, we give the same proof as\n  that of~\\cite[Lemma 3.10]{MR3689384}.\n  Let \\(\\bm{e}\\) be the exponentially distributed with parameter\n  \\(1\\) and \\(\\bm{e}_q \\coloneqq \\bm{e}\/q\\) for \\(q>0\\).\n  Then we already have~\\eqref{eq:h_b-eq} and~\\eqref{eq:h_b-lower}.\n  Letting \\(q\\to 0+\\) in~\\eqref{eq:h_b-lower}, we obtain\n  \\begin{align}\n    \\liminf_{x\\to\\infty}h^B(x) \\ge \\kappa^{-1}.\n  \\end{align}\n  On the other hand, it holds that\n  \\begin{align}\n    h^B(x) \\le \\ensuremath{\\mathbb{P}}\\sbra{L_{\\bm{e}_q}} + \\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}; T_x>\\bm{e}_q}\n    = r_q(0) + \\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}; T_x>\\bm{e}_q}.\n  \\end{align}\n  Letting \\(q\\to 0+\\), we have\n  \\(h^B(x) \\le \\kappa^{-1}\\).\n  Therefore we obtain the desired result.\n\\end{proof}\n\n\\begin{Thm}\\label{Thm:n-two-hitting-time-trans}\n  For \\(a \\in \\ensuremath{\\mathbb{R}} \\setminus \\{0\\}\\), it holds that\n  \\begin{align}\n    n(T_a < T_0 < \\infty) = \\frac{1-\\kappa h^B(a)}{h^B(a)}, \\quad\n    \\text{and} \\quad\n    n(T_a < T_0) = \\frac{1-\\kappa h(-a)}{h^B(a)}.\n  \\end{align}\n\\end{Thm}\n\\begin{proof}\n  For \\(l > 0\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}(L_{T_a}>l) = \\ensuremath{\\mathbb{P}}(T_a > \\eta_l) =\n    \\ensuremath{\\mathbb{P}}(\\sigma_{\\cbra{T_0 =\\infty}\\cup \\cbra{T_a<T_0<\\infty}}> l),\n  \\end{align}\n  where \\(\\sigma_A = \\inf\\cbra{l \\colon e_l \\in A}\\)\n  for \\(A \\subset \\ensuremath{\\mathcal{D}}\\).\n  Since \\(\\sigma_A\\) is the hitting time\n  of the set \\(A\\)\n  for the killed\n  Poisson point process \\(((l, e_l), l\\ge 0)\\),\n  we have\n  \\begin{align}\\label{eq:L_T_a-exp-distrib-trans}\n    \\ensuremath{\\mathbb{P}}(L_{T_a}>l)\n    = \\mathrm{e}^{-l\\cbra{n(T_0=\\infty)+n(T_a<T_0<\\infty)}}\n    = \\mathrm{e}^{-l\\cbra{\\kappa +n(T_a<T_0<\\infty)}}.\n  \\end{align}\n  For more details, see, e.g.,~\\cite[Lemma 6.17]{MR3155252}.\n  In particular, \\(L_{T_a}\\) is exponentially distributed.\n  On the other hand, we know \\(h^B(a) = \\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}\\)\n  by Lemma~\\ref{Lem:h^B-hitting-h}.\n  Hence we obtain\n  \\begin{align}\n    n(T_a < T_0 < \\infty) = \\frac{1}{h^B(a)} - \\kappa.\n    \\label{eq:n-T_a-T_0-inf-trans}\n  \\end{align}\n  We use the strong Markov property of the excursion measure \\(n\\)\n  (see, e.g.,~\\cite[Theorem III.3.28]{MR1138461})\n  to obtain\n  \\begin{align}\n    n(T_a < T_0 < \\infty)\n    = n(\\ensuremath{\\mathbb{P}}_a(T_0 <\\infty); T_a<T_0)\n    = \\rbra{1-\\kappa h(a)} n(T_a<T_0).\n    \\label{eq:n-T_a-T_0}\n  \\end{align}\n  It follows from~\\eqref{eq:n-T_a-T_0-inf} and~\\eqref{eq:n-T_a-T_0}\n  that\n  \\begin{align}\n    n(T_a<T_0)\n    = \\frac{1-\\kappa h^B(a)}{h^B(a)(1-\\kappa h(a))}\n    = \\frac{1-\\kappa h(-a)}{h^B(a)}.\n  \\end{align}\n  Hence the proof is complete.\n\\end{proof}\n\n\\subsection{Penalization result in the transient case}\nLet \\(f\\) be a non-negative\nfunction on \\([0,\\infty)\\)\nwhich satisfies\n\\(\\int_0^\\infty \\mathrm{e}^{-\\kappa u}f(u)\\, \\mathrm{d} u <\\infty\\).\nFor \\(x \\in \\mathbb{R}\\), we introduce the process given by\n\\begin{align}\\label{eq:def-mart-trans}\n  M_t= M_t^{(f)}= h(X_t)f(L_t)+(1-\\kappa h(X_t))\n  \\int_0^\\infty\n  \\mathrm{e}^{-\\kappa u} f(L_t+u)\\,\\mathrm{d} u.\n\\end{align}\nThen we can show that \\((M_t,t\\ge 0)\\) is a non-negative martingale.\n\\begin{Thm}\\label{Thm:transient-infty-mart}\n  Let \\(x \\in \\mathbb{R}\\) and let \\(f\\) be a non-negative\n  function on \\([0,\\infty)\\)\n  which satisfies\n  \\(\\int_0^\\infty \\mathrm{e}^{-\\kappa u}f(u)\\, \\mathrm{d} u <\\infty\\).\n  Then it holds that\n  \\begin{align}\\label{eq:L-inf-M-trans}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_\\infty)}\n    = \\kappa M_0,\\quad \\ensuremath{\\mathbb{P}}_x\\text{-a.s.,}\n  \\end{align}\n  and \\((M_t,t\\ge 0)\\) is a non-negative \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{Thm}\n\\begin{proof}\n  By the same discussion, Lemma~\\ref{Lem:expect-L_e_q} also holds\n  in the transient case.\n  Suppose first that \\(f\\) is bounded.\n  We write \\(g = \\sup\\{t\\colon X_t=0\\}\\).\n  Since \\(X\\) is transient, \\(\\ensuremath{\\mathbb{P}}_x(g < \\infty)=1\\).\n  We see that \\(f(L_{\\bm{e}_q})\\to f(L_\\infty)\\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q\\to 0+\\); in fact,\n  for almost every sample path,\n \\(L_{\\bm{e}_q} = L_g = L_\\infty\\) for small \\(q>0\\)\n (Here we do not need continuity of \\(f\\)).\n  Hence, by the dominated convergence theorem, we obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})} \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_\\infty)},\n    \\qquad \\text{as \\(q \\to 0+\\).}\n  \\end{align}\n  On the other hand, by the monotone convergence theorem, we obtain\n  \\begin{align}\n    \\int_0^\\infty \\mathrm{e}^{-u\/{r_q(0)}}f(u) \\, \\mathrm{d} u\n    \\longrightarrow \\int_0^\\infty \\mathrm{e}^{-\\kappa u} f(u)\\, \\mathrm{d} u,\n    \\qquad  \\text{as } q \\to {0+}.\n  \\end{align}\n  Hence~\\eqref{eq:L-inf-M-trans} follows by letting \\(q \\to 0+\\)\n  in~\\eqref{eq:lem-expect-L_e_q}.\n  To remove the boundedness assumption of \\(f\\),\n  we consider \\(f \\wedge n\\) and then let \\(n \\to \\infty\\) in~\\eqref{eq:L-inf-M-trans}.\n  Moreover, by the Markov property and the additivity of \\(L\\), we have,\n  for \\(0<s<t\\),\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t|\\ensuremath{\\mathcal{F}}_s}\n     & =\\kappa^{-1}\n    \\ensuremath{\\mathbb{P}}_x\\sbra[\\big]{\\widetilde{\\ensuremath{\\mathbb{P}}}_{X_t}\\sbra{f(\\widetilde{L}_\\infty+L_t)}|\\ensuremath{\\mathcal{F}}_s}\n    \\\\\n     & =\\kappa^{-1}\\ensuremath{\\mathbb{P}}_x\\sbra[\\big]{{\\ensuremath{\\mathbb{P}}}_{x}\\sbra{f({L}_\\infty)|\\ensuremath{\\mathcal{F}}_t}|\\ensuremath{\\mathcal{F}}_s}\n    \\\\\n     & = \\kappa^{-1}{\\ensuremath{\\mathbb{P}}}_{x}\\sbra{f({L}_\\infty)|\\ensuremath{\\mathcal{F}}_s}\n    \\\\\n     & = \\kappa^{-1}\\widetilde{\\ensuremath{\\mathbb{P}}}_{X_s}\\sbra{f(\\widetilde{L}_\\infty+L_s)}    \\\\\n     & = M_s.\n  \\end{align}\n  This means that \\((M_t,t\\ge 0)\\) is a non-negative \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{proof}\n\n\\begin{Thm}\\label{Thm:penal-trans}\n  Suppose that\n  the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\n  Let \\(f\\) be a bounded non-negative function\n  and let \\(\\tau\\) be a random clock. Define\n  \\begin{align}\n    M_t^\\tau & = \\kappa^{-1}\\ensuremath{\\mathbb{P}}_x\\sbra{ f(L_\\tau)|\\ensuremath{\\mathcal{F}}_t}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    M_t^\\tau \\longrightarrow M_t ,\n    \\quad \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(\\tau\\to\\infty\\).}\n  \\end{align}\n  Consequently, if \\(M_0>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\\label{eq:penalized-meas-trans}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\tau})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\tau})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t}{M_0}},\n    \\qquad \\text{as \\(\\tau \\to \\infty\\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\n\\begin{proof}\n  We have\n  \\(f(L_\\tau)\\to f(L_\\infty)\\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\n  as \\(\\tau\\to\\infty\\); in fact,\n  \\(L_\\tau=L_g=L_\\infty\\) for large \\(\\tau\\).\n  In addition, since \\(f\\) is bounded,\n  we may apply the dominated convergence theorem to obtain\n  \\(f(L_\\tau)\\to f(L_\\infty)\\), in\n  \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(\\tau\\to\\infty\\).\n  This implies that \\(M_t^\\tau \\to M_t\\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and in\n  \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(\\tau\\to\\infty\\).\n\\end{proof}\n\\begin{Rem}\n  If \\(\\tau\\) is exponential clock,\n  hitting clock, two-point hitting time clock or\n  inverse local time clock, Theorem~\\ref{Thm:penal-trans}\n  also holds under the assumption that\n  \\(f\\) is a non-negative\n  function which satisfies\n  \\(\\int_0^\\infty \\mathrm{e}^{-\\kappa u}f(u)\\, \\mathrm{d} u <\\infty\\).\n\\end{Rem}\n\nSince we have \\(M_t = \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_\\infty)|\\ensuremath{\\mathcal{F}}_t}\\),\nwe see that the penalized measure \\(\\ensuremath{\\mathbb{Q}}^f_x\\) can be represented as\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}^f_x = \\frac{f(L_\\infty)}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_\\infty)}} \\cdot \\ensuremath{\\mathbb{P}}_x,\n\\end{align}\nwhich shows that \\(\\ensuremath{\\mathbb{Q}}^f_x\\) is absolutely continuous with respect to \\(\\ensuremath{\\mathbb{P}}_x\\).\n\n\\section{Appendix: Martingale property of \\(X_t f(L_t)\\)}\\label{Sec:mart}\nIn Remark~\\ref{Rem:martingale}, we have shown that\n\\((X_t f(L_t),t\\ge 0)\\) is a \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale\nfor \\(f\\in\\ensuremath{\\mathcal{L}}^1_{+}\\) under the condition that \\(m^2<\\infty\\).\nLet us remove the additional assumption \\(m^2<\\infty\\).\nIn this section, we assume \\(X\\) is either recurrent or transient,\nand assume the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular}.\n\\begin{Thm}\\label{Thm:mart-XfL}\n  Suppose the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\n  Suppose, in addition, that \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_1}}<\\infty\\) and \\(\\ensuremath{\\mathbb{P}}[X_1] = 0\\).\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{\\bm{e}_q}}}<\\infty\\) and\n          \\(n\\sbra{\\abs{X_{\\bm{e}_q}};T_0>\\bm{e}_q}<\\infty\\)\n          for all \\(q>0\\), and \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}<\\infty\\)\n          and \\(n\\sbra{\\abs{X_t};T_0>t}<\\infty\\)\n          for all \\(t>0\\);\\label{Thm-item:pn-integ}\n    \\item  \\(\\ensuremath{\\mathbb{P}}_x\\sbra{X_t; T_0>t} = x\\), for all \\(t>0\\)\n          and \\(x\\in\\ensuremath{\\mathbb{R}}\\);\\label{Thm-item:p-kill-expect}\n    \\item  \\(n\\sbra{X_t;T_0>t}=0\\), for all \\(t>0\\);\\label{Thm-item:n-expect}\n    \\item  \\((X_t f(L_t), t\\ge 0 )\\) is a\n          \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale\n          for \\(x \\in \\ensuremath{\\mathbb{R}}\\) and all bounded measurable functions\n          \\(f\\).\\label{Thm-item:mart-XfL}\n  \\end{enumerate}\n\\end{Thm}\n\\begin{proof}\n  \\noindent (\\ref{Thm-item:pn-integ})\n  We have \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{t+s}}}\\le\n  \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}+\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{t+s}-X_t}}\n  =\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}+\n  \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{s}}}\\),\n  which implies that the function \\(t\\mapsto \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}\\) is subadditive.\n  Hence, for any \\(k\\in \\ensuremath{\\mathbb{N}}\\),\n  we have\n  \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_k}}\\le k\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_1}}<\\infty\\).\n  For \\(t>0\\), it is known that\n  \\begin{align}\\label{eq:sup-above}\n    \\ensuremath{\\mathbb{P}}\\sbra*{\\sup_{0\\le s\\le t}\\abs{X_s}}\\le 8\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}};\n  \\end{align}\n  see Doob~\\cite[Theorem VII.5.1]{MR0058896} and\n  Sato~\\cite[Theorem 25.18 and Remark 25.19]{MR1739520}\n  for the proof. Hence we have \\(\\sup_{0\\le t\\le k} \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}<\\infty\\)\n  for all \\(k\\in\\ensuremath{\\mathbb{N}}\\).\n  In particular, we obtain \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}<\\infty\\) for all \\(t>0\\).\n  Again by the subadditivity of \\(t\\mapsto \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}\\), we have\n  \\begin{align}\n    \\lim_{t\\to\\infty}\\frac{\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}}{t}=\n    \\inf_{t>0}\\frac{\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}}{t}\n    \\le\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_1}}<\\infty.\n  \\end{align}\n  Thus there exist constants \\(C,C'>0\\) such that \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}\\le C+C't\\)\n  for all \\(t>0\\).\n  In particular, we obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{\\bm{e}_q}}}\\le\n    q\\int_0^\\infty (C+C't)\\mathrm{e}^{-qt} \\,\\mathrm{d} t<\\infty,\n    \\quad \\text{for all \\(q>0\\).}\n  \\end{align}\n  By Lemma~\\ref{Lem:compensation-formula}, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{\\bm{e}_q}}}\n     & = \\ensuremath{\\mathbb{P}}\\sbra*{\\sum_{u\\in D}\\int_{\\eta_{u-}}^{\\eta_u}q\\mathrm{e}^{-qt} \\abs{X_t}\n    \\, \\mathrm{d} t}                                                                   \\\\\n     & = \\ensuremath{\\mathbb{P}} \\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_u \\, q\\mathrm{e}^{-qu}\n    \\int_0^{\\widetilde{T}_0} \\mathrm{d} t\\, \\mathrm{e}^{-qt}\\abs{\\widetilde{X}_t}}            \\\\\n     & = \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty \\mathrm{d} L_u \\, \\mathrm{e}^{-qu}}\n    n\\sbra{\\abs{X_{\\bm{e}_q}}; T_0>\\bm{e}_q}\n    \\\\\n     & = r_q(0) n\\sbra{\\abs{X_{\\bm{e}_q}}; T_0>\\bm{e}_q}.\n  \\end{align}\n  Hence we have \\(n\\sbra{\\abs{X_{\\bm{e}_q}}; T_0>\\bm{e}_q}<\\infty\\) for all \\(q>0\\)\n  and this implies that \\(n\\sbra{\\abs{X_{t}};T_0>t}<\\infty\\)\n  for almost all \\(t>0\\).\n  For any \\(t>0\\), we can take \\(0<s<t\\) such that\n  \\(n\\sbra{\\abs{X_s};T_0>s}<\\infty\\).\n  Then it follows from the Markov property of the excursion measure \\(n\\)\n  that\n  \\begin{align}\n    n\\sbra{\\abs{X_t};T_0>t}\n     & = n\\sbra{\\ensuremath{\\mathbb{P}}_{X_s}\\sbra{\\abs{X_{t-s}}; T_0>t-s}; T_0>s}        \\\\\n     & \\le  n\\sbra{\\ensuremath{\\mathbb{P}}_{X_s}\\sbra{\\abs{X_{t-s}}}; T_0>s}              \\\\\n     & \\le n\\sbra{\\abs{X_s}+\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{t-s}}}; T_0>s}<\\infty.\n  \\end{align}\n  Hence we obtain \\(n\\sbra{\\abs{X_t};T_0>t}<\\infty\\) for all \\(t>0\\).\n\n  \\noindent (\\ref{Thm-item:p-kill-expect})\n  By the Markov property, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{X_t}\n    = \\ensuremath{\\mathbb{P}}_x\\sbra{X_t; T_0>t}\n    +\\int_{[0,t]}\\ensuremath{\\mathbb{P}}_x(T_0\\in\\mathrm{d} s)\\ensuremath{\\mathbb{P}}\\sbra{X_{t-s}}.\n  \\end{align}\n  Since \\(\\ensuremath{\\mathbb{P}}_x\\sbra{X_t}=x\\) for all \\(t\\ge 0\\) and \\(x\\in\\ensuremath{\\mathbb{R}}\\), we obtain\n  \\(\\ensuremath{\\mathbb{P}}_x\\sbra{X_t; T_0>t}=x \\) for all \\(t>0\\).\n\n  \\noindent (\\ref{Thm-item:n-expect})\n  By Lemma~\\ref{Lem:compensation-formula},\n  we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q} f(L_{\\bm{e}_q})}\n     & = \\ensuremath{\\mathbb{P}} \\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_u \\, q\\mathrm{e}^{-qu} f(u)\n    \\int_0^{\\widetilde{T}_0} \\mathrm{d} t\\, \\mathrm{e}^{-qt}\\widetilde{X}_t}  \\\\\n     & = \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty \\mathrm{d} L_u \\, q \\mathrm{e}^{-qu} f(u)}\n    \\int_0^\\infty \\mathrm{d} t \\, \\mathrm{e}^{-qt} n\\sbra{X_t;T_0>t}.\n    \\label{eq:X_eqL_eq}\n  \\end{align}\n  Here in the second equality, we use Fubini's theorem.\n  If we take \\(f \\equiv 1\\),\n  then~\\eqref{eq:X_eqL_eq} becomes\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q}}\n    = qr_q(0)\n    \\int_0^\\infty \\mathrm{d} t \\, \\mathrm{e}^{-qt} n\\sbra{X_t;T_0>t}.\n  \\end{align}\n  Since \\(\\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q}} = 0\\) for all \\(q > 0\\),\n  we obtain \\(n\\sbra{X_t;T_0>t} = 0\\)\n  for almost all \\(t>0\\).\n  By the Markov property of the excursion measure \\(n\\),\n  we have, for \\(0<s<t\\),\n  \\begin{align}\n    n\\sbra{X_t;T_0>t} = n\\sbra{\\ensuremath{\\mathbb{P}}_{X_s}\\sbra{X_{t-s}; T_0>t-s};T_0>s}\n    = n\\sbra{X_s;T_0>s},\n  \\end{align}\n  which implies that \\(n\\sbra{X_t;T_0>t}\\) is constant in \\(t>0\\).\n  Thus we obtain \\(n\\sbra{X_t;T_0>t}=0\\) for all \\(t>0\\).\n\n  \\noindent (\\ref{Thm-item:mart-XfL})\n  By~(\\ref{Thm-item:n-expect}) of\n  Theorem~\\ref{Thm:mart-XfL} and by~\\eqref{eq:X_eqL_eq},\n  we have\n  \\(\\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q}f(L_{\\bm{e}_q})} = 0\\).\n  Hence, by the Markov property and~(\\ref{Thm-item:p-kill-expect}) of\n  Theorem~\\ref{Thm:mart-XfL},\n  it holds that,\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{X_{\\bm{e}_q}f(L_{\\bm{e}_q})}\n     & =\n    \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^{T_0}q\\mathrm{e}^{-qt} X_t f(0)\\, \\mathrm{d} t}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_0}}\n    \\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q} f(L_{\\bm{e}_q})} \\\\\n     & =\n    f(0) \\int_0^\\infty q\\mathrm{e}^{-qt} \\ensuremath{\\mathbb{P}}_x\\sbra{X_t ;T_0>t}\n    \\, \\mathrm{d} t                                 \\\\\n     & =x f(0).\n    \\label{eq:x-X_eqL_eq}\n  \\end{align}\n  Thus we obtain\n  \\begin{align}\\label{eq:XL-a.e.}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{X_t f(L_{t})} = xf(0) \\quad\n    \\text{ for almost all \\(t > 0\\).}\n  \\end{align}\n  We show the function \\(t\\mapsto \\ensuremath{\\mathbb{P}}_x\\sbra{X_t f(L_{t})}\\)\n  is right-continuous for \\(t>0\\).\n  Fix \\(t>0\\).\n  Since \\(X_t\\ne 0\\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.,\n  we see that, for almost every sample path,\n  we can choose small \\(\\delta>0\\) such that\n  \\(L_{t+\\delta}=L_t\\). Since \\(t\\mapsto X_t\\) is right-continuous,\n  we have\n  \\(\\lim_{s\\to 0+}X_{t+s}f(L_{t+s}) = X_t f(L_t) \\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.,\n  where we do not require continuity of \\(f\\).\n  For \\(0<s<1\\), it holds that\n  \\(\\abs{X_{t+s}f(L_{t+s})}\\le \\sup_{0\\le t'\\le t+1}\n  \\abs{X_{t'}}\\sup_{u\\in\\ensuremath{\\mathbb{R}}}\\abs{f(u)}\\).\n  By~\\eqref{eq:sup-above}, we may apply the dominated convergence theorem to deduce\n  that\n  \\(t\\mapsto\\ensuremath{\\mathbb{P}}_x\\sbra{X_t f(L_{t})} \\) is right-continuous.\n  This and~\\eqref{eq:XL-a.e.} imply that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{X_t f(L_{t})} = xf(0) \\quad \\text{for all \\(t\\ge 0\\)}.\n  \\end{align}\n  By the Markov property and the additivity property of \\(L\\),\n  the process \\((X_t f(L_t), t\\ge 0)\\) is a\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{proof}\nBy Theorems~\\ref{Thm:martingale} and~\\ref{Thm:mart-XfL}, we obtain the following:\n\\begin{Cor}\n  Suppose that the assumptions of Theorem~\\ref{Thm:mart-XfL} are satisfied\n  and that \\(X\\) is recurrent.\n  For measurable function \\(f_1\\) and locally integrable function \\(f_2\\), define\n  \\begin{align}\n    F(X_t,L_t)=X_t f_1(L_t) + h(X_t)f_2(L_t) -\\int_0^{L_t} f_2(u)\\,\\mathrm{d} u.\n  \\end{align}\n  Then the following assertions hold.\n  \\begin{enumerate}\n    \\item If \\(f_1\\) is bounded and \\(f_2\\) is integrable, then\n          \\((F(X_t,L_t),t\\ge 0)\\) is a \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n    \\item If \\(f_1\\) is locally bounded and \\(f_2\\) is locally integrable,\n          then \\((F(X_t,L_t),t\\ge 0)\\) is a local \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  \\end{enumerate}\n\\end{Cor}\n\n\\begin{Rem}\n  For the Brownian motion \\(B\\) and its local time \\(L\\) at \\(0\\),\n  Fitzsimmons--Wroblewski~\\cite{MR2759773}\n  proved that any local martingale of the\n  form \\((F(B_t, L_t), t\\ge 0)\\)\n  is given by the following form:\n  \\begin{align}\\label{eq:F-mart}\n    F(B_t,L_t) & = F(0,0)+B_t f_1(L_t) +\\abs{B_t}f_2(L_t) -\\int_0^{L_t} f_2(u)\\, \\mathrm{d} u,\n  \\end{align}\n  where \\(f_1\\) and \\(f_2\\) are locally integrable functions.\n\\end{Rem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nData privacy protection is becoming increasingly important; not only are data breaches gaining public attention, but there are also data protection initiatives and data privacy laws in place, such as the General Data Protection Regulation (GDPR)  and the California Consumer Privacy Act (CCPA).\nDeep learning, on the other hand, poses a significant risk of data privacy leakage since the model embeds information about the training data. For example, both\n \\cite{deep_leakage} and \\cite{IDLG}  provide paradigms for reconstructing training examples from published models. As a result, privacy-preserving algorithms are becoming increasingly important for preserving data privacy. \n \n Differential privacy (DP) is a provable and quantifiable method for privacy protection \\citep{DBLP:conf\/icalp\/Dwork06}, which guarantees it nearly impossible for the adversary to differentiate  from two {\\it neighboring data sets}.  { However, applying DP to maintain the privacy of the training data in deep learning is extremely difficult  due to the thousands to millions of repetitions\\footnote{Around 300,000 steps is needed to train large models like GPT3.} of privacy budget cost in stochastic gradient descent (SGD) iterations, resulting in a very high additive noise power that completely degrades the learning process within a reasonable privacy preserving region.\n}\n\nThe first DP-SGD with a reasonable level of accuracy is proposed by \\cite{DBLP:conf\/ccs\/AbadiCGMMT016}. The calibrated noise added to the clipped gradient is much smaller than all previous methods due to their moments-accountant method for tight privacy accounting.\nEven so, the accuracy drop when compared to the non-DP model is still significant. In the MNIST dataset, for example, the moment-accountant based DP-SGD sacrifices accuracy by about $4.5\\%$ with the privacy protection level $\\epsilon=1.34$~\\citep{bu2019deep}. Performance will continue to suffer as the level of privacy protection increases. This motivates us to narrow the performance gap in this paper without compromising privacy.\n\nTo begin, motivated by the instability and even ramp-up of DP-SGD updates, as shown later in Fig.~\\ref{fig:Gradient_Norm}, we investigate how to perform a tight DP accounting to enable proactive DP budget allocation for each training step to avoid unstable updates.\nSecond, under this new privacy accounting framework, we propose instances of algorithms for proactive privacy budget allocation, such as the sensitivity-decay method, the growing-$\\mu_t$ method, and a hybrid of the two known as dynamic DP-SGD.\nThird, we thoroughly validate the performance of dynamic DP-SGD for a variety of deep learning tasks, demonstrating  consistently significant accuracy improvements over existing methods while preserving privacy. The technical contributions are summarized below.\n\n\n\n\\begin{itemize}\n\\item We broaden Gaussian DP's central limit theorem (CLT) \nto obtain a tight bound for the  composition of  Gaussian mechanisms with different privacy parameters  considering subsampling amplification.\nThe closed form of the CLT result facilitates the proactive  calibration of different noise powers and clipping values at each training step.\n\n\\item \nWe investigate how to proactively allocate the DP cost based on the CLT result in such a way that the gradient updates are stabilized and thus model performance is improved. \nWe propose the  {\\it sensitivity decay} and {\\it growing-$\\mu_t$} methods, and demonstrate that combining these two methods results in the best performance, which is referred to as  {\\it dynamic DP-SGD}.\n\n\n\n\n\n\\item \nWe perform a series of experiments and ablation studies on a variety of neural network tasks, including image classification, natural language processing, and federated learning, and  show that  the proposed dynamic DP-SGD effectively stabilizes gradient updates and consistently outperforms existing methods. With a strong privacy guarantee, i.e., at $\\epsilon = 1.2 $, dynamic DP-SGD has a performance loss of only $2.49\\%$ when compared to the none-DP version on MNIST dataset, and the loss is reduced to $1.72\\%$ in the federated learning setting for a stronger privacy guarantee, i.e., $\\epsilon=1$ due to a larger DP amplification. \n\\end{itemize}\n\n\n\\section{Related Work}\nThe following section summarizes related work on algorithm design and DP accounting for DP-SGD.\n\n\\noindent\\textbf{{DP-SGD Algorithm}}\nTo improve the model's accuracy, previous work has concentrated on designing variations of DP-SGD  by estimating the clipping bound and minimizing the bias introduced by gradient clipping. More precisely, \\cite{DBLP:conf\/ccs\/AbadiCGMMT016} propose norm clipping and per-layer clipping, both of which select clipping values based on gradient differences between different layers.\n\\cite{AdaClip} pioneer AdaClip, a coordinate-wise clipping method that significantly reduces the total amount of noise required.\n\\cite{Quantile} introduce  gradient clipping  based on  the quantile statistics of the gradient, which requires additional DP cost to protect those quantiles.\nRecently, \\cite{DBLP:conf\/nips\/ChenWH20} analyze the bias introduced by the gradient clipping operation and propose a method for reducing the bias error by first adding noise before clipping.\nIt's worth noting that the proposed dynamic DP-SGD in this paper is compatible with the methods mentioned above and can be used in tandem to investigate accuracy improvement.\n\nFurthermore,\n\\cite{yu2019differentially} provide a means of reducing noise variance during the DP-SGD process, thereby improving model performance; and  \\cite{zhang2021adaptive} analyze the DP cost for the same method using the z-CDP privacy accounting.\nDue to the loose DP accountings, these methods have a large performance gap in the high privacy guaranteed region. As demonstrated later in the experiments, the proposed dynamic DP-SGD improves these results significantly due to the dynamic clipping operation and tight DP composition. \n\n\n\\noindent\\textbf{Privacy Accounting}\nEach step in the DP-SGD process depletes the total privacy budget.\nFollowing $T$ steps of training, DP accounting necessitates the composition of these mechanisms in order to calculate the total privacy cost in terms of $(\\epsilon, \\delta)$.\nSimple composition~\\citep{dwork2006our, dwork2009differential} with $\\epsilon=\\mathcal O(T)$ and advance composition~\\citep{dwork2010boosting} with $\\epsilon=\\mathcal O(\\sqrt{T})$ have been proposed in the literature.\n If $M_{1}, M_{2}, \\ldots, M_{T}$ are distinct DP mechanisms such that $M_{i}$ is $\\left(\\epsilon_{i}, \\delta_{i}\\right)$-DP, then it is shown by \\cite{murtagh2016complexity} that computing the exact DP guarantees for the composition  $M=M_{1} \\circ M_{2} \\circ \\cdots \\circ M_{T}$ is \\#P-complete.\n\\cite{DBLP:conf\/ccs\/AbadiCGMMT016} propose the moment accountant of DP-SGD. By establishing a privacy cost function, the moment accountant imposes a tight constraint on the estimation of privacy loss, calibrating the  noise power down to a practicable level for the first time in the privacy preserving deep learning context.\nFrom a broader perspective, \\cite{rdp} proposes to use R\\'enyi divergence to determine the distance between the outputs of two adjacent datasets, thereby establishing the RDP concept, and \\cite{balle2020privacy} provide an advance conversion from RDP to $(\\epsilon,\\delta)$-DP.\nDue to the inherent use of subsampling in training neural networks, the DP composition with subsampling DP amplification is studied in detail by \\cite{wang2019subsampled, balle2020privacy}, which reduce the noise power  for the privacy preserving.\n\\cite{dong2019gaussian} recently propose the concept of f-DP to quantify the privacy cost from a hypothesis testing perspective, with Gaussian DP (GDP) as a major application.\nWhile GDP allows for a tight composition, computing the exact composition of the Gaussian mechanism with subsampling amplification is computationally challenging. It is also demonstrated that a computationally efficient central limit theorem (CLT) can approximate the composition of multiple identical DP mechanisms.\nVery recently, \\citet{gopi2021numerical, zhu2021optimal} propose optimal composition of DP mechanisms via numerical methods, but it is unclear how to calibrate different noise power and clipping value at each step as in the proposed dynamic DP-SGD.\n\n\n \n \n\n \n\n \n\n\\section{An Inspiring Case of the Unstable DP-SGD}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = 0.33\\textwidth]{gradient_norm_3_lines2.jpg}\n    \\caption{Experiments on MNIST with DP budget $(\\epsilon,\\delta)=(0.4,10^{-5})$ after $5\\times 10^3$ steps for both DP-SGD and proposed dynamic DP-SGD. In comparison to SGD without DP protection, DP-SGD is unstable and has a ramp-up period. By contrast, the gradient norm is stabilized by the proposed dynamic DP-SGD.  }\n    \\label{fig:Gradient_Norm}\n\\end{figure}\n\nDP defines an upper bound of the privacy level by computing a pair of $(\\epsilon, \\delta(\\epsilon))$ on the privacy budget.\nLower $\\epsilon$ values indicate better privacy protection but also potentially the less useful the protected algorithm is.\nThe value $\\delta$ can be interpreted as the probability of failing to achieve DP.\nWith the notion of the neighboring data sets, i.e., $X\\sim X^{\\prime}$, which differs by one {\\it data record},\nthe formal $(\\epsilon, \\delta)$-DP definition is given below~\\citep{DBLP:conf\/icalp\/Dwork06}.\n\\begin{definition} (($\\epsilon, \\delta(\\epsilon)$)-DP Profile)\n A randomized algorithm $M(\\cdot)$ gives $(\\epsilon, \\delta(\\epsilon))$-differential privacy if for any pair of neighboring datasets $X\\sim X^{\\prime}$ and any event $E$,\n$$\n\\mathbb{P}(M(X) \\in E) \\leqslant \\mathrm{e}^{\\epsilon} \\mathbb{P}\\left(M\\left(X^{\\prime}\\right) \\in E\\right)+\\delta,\n$$\n\\end{definition}\nwhere the probability $\\mathbb{P}(\\cdot)$ is taken over the randomness of $M$, and $\\epsilon\n\\geq 0$.\nWhen $\\delta = 0$, the algorithm is $\\epsilon$-DP.\nIntuitively, this means that we can't tell whether $M$ was run on $X$ or $X^{\\prime}$ based on the results.\nAs a result, an adversary will almost never infer the existence of any specific data record in the input data set.\n\nIn a Gaussian mechanism,  let $Y$ be the random variable following  Gaussian distribution with  $Y\\sim \\mathcal N\\left(0, \\sigma^2I_d\\right)$ and $f: X^{n} \\rightarrow \\mathbb{R}^{d}$.\nThe Gaussian mechanism $ M(X)=f(X)+ Y$ follows the  DP profile~\\citep{wang2019subsampled}:\n\\begin{equation}\n\\label{compute_mu}\n\\delta(\\epsilon ; \\mu)\n=\\Phi\\left(-\\frac{\\epsilon}{\\mu}+\\frac{\\mu}{2}\\right)-\\mathrm{e}^{\\epsilon} \\Phi\\left(-\\frac{\\epsilon}{\\mu}-\\frac{\\mu}{2}\\right),\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{mu-def}\n\\mu = \\frac{C}{\\sigma}\n\\end{equation}\nwith  $C$ the sensitivity of $f(X)$ and $\\Phi(\\cdot)$  the Gaussian cumulative distribution function. \nThe DP-SGD is proposed below based on the Gaussian mechanism.\nLet the model parameters be denoted by $\\theta$. The gradient computed by a data sample $x$ in the SGD is given by  \n$\ng_x \\triangleq \\frac{\\partial f_x}{\\partial \\theta},\n$ where $f_x$ is the corresponding loss function.\nTo calibrate the noise  required for DP, a paradigm is to first clip the $\\ell_2$-norm of the gradient, i.e.,\n\\begin{equation}\n\\label{clipping}\n  \\widetilde g_x \\triangleq \n\\operatorname{CL}\\left({g_x}; C\\right) \n\\triangleq \ng_x\\cdot\\min\\left(1, \\frac{C}{\\left\\|{g_x} \\right\\|}\\right),\n\\end{equation}\nand in the subsequent, to add the calibrated noise $\\xi_t\\sim \\mathcal N(0,\\sigma^2)$ with $\\sigma = \\frac{C}{\\mu}$. Thereby, the DP-SGD updates at the $t$-th step is given by~\\citep{DBLP:conf\/ccs\/AbadiCGMMT016}:\n\\begin{align}\n\\label{dpsgd}\n\\text{DP-SGD:} \\quad\n\\begin{split}\n  \\theta_t = \\theta_{t-1} - \\eta \n    \\frac{1}{\\left|X_{t}\\right|} \\left( \n    \\sum_{x\\in X_{ t}}\\widetilde g_x\n    + \\xi_{t}\\right), \\quad t\\in[T].\n\\end{split}\n\\end{align}\nBecause of DP's post-processing property, protecting gradients provides the same level of privacy protection on the output model.\nA closer examination of the DP-SGD process in (\\ref{dpsgd}) reveals that the clipping-per-sample operator introduces biases to the original unbiased gradient estimate in the SGD updates if $C$ is not large enough. It is possible to have an unbiased gradient estimate by increasing $C$ if $C$ is greater than any $\\frac{\\partial f_x}{\\partial \\theta}$. It does, however, result in an over-calibrated noise power.\n\n\\textbf{Motivations of this work}:\nIn standard SGD updates, the gradient norm is reduced to a very small value. However, because the DP-SGD is followed by evenly consuming the privacy budget, the ratio of noise power to the true gradient norm in~Eqn. (\\ref{dpsgd}) would continue to rise, resulting in unstable updates.\nThis hypothesis is supported by Fig.~\\ref{fig:Gradient_Norm}, which displays the average coordinate gradient norm for all the data records in each step during the iterative updates.\nIt is worth noting that the DP-SGD gradient norm has a ramp-up period. This phenomenon  also appeared in the experiments in~\\citep{Quantile}. The observation motivates us to investigate a dynamic DP-SGD to reduce both the clipping value and the noise power in order to stabilize the updates. As a result, we can control the rate at which the privacy budget consumed.\nDespite the idea of a dynamic DP-SGD, existing DP accounting methodologies are inadequate to facilitate the study on under what criterior to allocate the privacy budget. As a result, in the following section, we first present an extended CLT of DP accounting  based on Gaussian DP (GDP)~\\citep{dong2019gaussian} and then demonstrate how to achieve dynamic DP-SGD with the extended CLT of GDP to improve model accuracy.\n\n\n\n\n\n\\section{Dynamic DP-SGD}\n\nThe DP parameter $(\\epsilon, \\delta)$ must be carefully determined, and existing theoretical analysis is frequently insufficient. In the following section, we work toward making DP machine learning practical by calibrating a reasonable noise power first providing refined privacy analyses and then designing a dynamic DP-SGD with a better convergence.\n\\subsection{Extended CLT for GDP}\n\\textbf{GDP Preliminary}: We first introduce some background about GDP.\nLet $P$ and $Q$ denote the distributions of $M(X)$ and $M\\left(X^{\\prime}\\right)$ with $X\\sim X^{\\prime}$, and let $\\phi$ be any (possibly randomized) rejection rule for testing $H_{0}: P$ against $H_{1}: Q$. With these in place, \\citet{dong2019gaussian} defines the trade-off function of $P$ and $Q$ as\n\\begin{equation}\n\\begin{aligned}\n\\text{T}(P, Q):[0,1] & \\mapsto[0,1] \\\\\n\\alpha & \\mapsto \\inf _{\\phi}\\left\\{1-\\mathbb{E}_{Q}[\\phi]: \\mathbb{E}_{P}[\\phi] \\leqslant \\alpha\\right\\}.\n\\end{aligned}\n\\end{equation}\nAbove,  $\\mathbb{E}_{P}[\\phi]$ and  $1-\\mathbb{E}_{Q}[\\phi]$  are type I and type II errors of the rejection rule  $\\phi$, respectively.\nIt is shown that $\\text{T}(P,Q)\\geq \\text{T}(\\mathcal N(0,1),\\mathcal N(\\mu,1))\\triangleq G_{\\mu}$, which is referred to as  $\\mu$-GDP.\n\nIn each step of the  DP-SGD in (\\ref{dpsgd}) with the Gaussian mechanism, it achieves $\\mu$-GDP with $\\mu = \\frac{C}{\\sigma}$.\nConsider the  sampling scheme $\\operatorname{PS}(X)$ that each individual data sample $(x, y)$ is subsampled independently with probability $p$ from the training set to construct $X_t$.\nIt is shown in \\citep{bu2019deep} that\ngiven two neighboring datasets $X$ and \n$X^{\\prime}$, \nif a randomized mechanism $\\mathcal M$ is $G_{\\mu}$-{DP},  then\n\\begin{equation}\n\\label{GDP-sampleing}\n\\text{T}\\left(\\mathcal M \\circ \\operatorname{PS} (X), \\mathcal M \\circ \\operatorname{PS}\\left(X^{\\prime}\\right)\\right) \\geqslant p G_{\\mu}+(1-p) \\mathrm{Id},\n\\end{equation}\nwhere $\\operatorname{Id}(x) = 1 - x$.\nThen after a large enough $T$ steps, a Berry-Esseen style CLT result is shown by \\cite{bu2019deep} that as $T\\to +\\infty$ and $p\\sqrt{T}\\to$ a constant,  the composition of the r.h.s. of (\\ref{GDP-sampleing}) converges to a $G_{\\mu_{\\text{tot}}}$-DP with\n\\begin{equation}\n\\label{CLT1}\n        \\mu_{\\text{tot}} = p\\sqrt{T(e^{\\mu^2}-1)}.\n\\end{equation}\n\n\\textbf{Extended CLT of GDP}: The preceding CLT result is based on the assumption that each step satisfies the same $G_{\\mu}$-DP, which restricts the possibility to calibrate noise for each individual step. Following that, in order to pave the way for DP accounting of a dynamic DP-SGD, we investigate the extended CLT in which each step $t$ has a different $G_{\\mu_t}$-DP.\nLet $\\mathcal M_A$ and $\\mathcal M_B$ denote the compositions of $T$ steps updates for the neighboring data sets $X$ and $X^{\\prime}$, respectively.\nAccording to~(\\ref{dpsgd}), each step consists of subsampling and local updates.\nThen, we express  $\\mathcal M_A$ and $\\mathcal M_B$ by\n$$\\mathcal M_A \\triangleq\nM_{T} \\circ \\operatorname{PS}_T (X)\\circ\\cdots\\circ\\mathcal M_{1} \\circ \\operatorname{PS}_1 (X), \\quad\\mathcal M_B \\triangleq\nM_{T} \\circ \\operatorname{PS}_T (X^{\\prime})\\circ\\cdots\\circ\\mathcal M_{1} \\circ \\operatorname{PS}_1 (X^{\\prime}).$$\nThe composition theorem in \\citep{bu2019deep} gives\n\\begin{eqnarray}\n\\label{T-Tradeoff}\n{\\text T}(\\mathcal M_A, \\mathcal M_B)\n\\geqslant\n\\otimes_{t=1}^T\n\\left(p \\cdot G_{\\mu_t}+(1-p) \\text { Id }\\right).\n\\end{eqnarray}\nThe symbol $\\otimes_{t=1}^T$ denotes the product of all the $T$ trade-off functions with the form $p \\cdot G_{\\mu_t}+(1-p) \\text { Id }$, which is far from analytically computable. \nIn the following theorem, we develop the CLT for the r.h.s of (\\ref{T-Tradeoff}) with the proof provided in Appendix~\\ref{apped:theorem1}.\n\\begin{theorem}\n\\label{MU-CLT}\nConsider a series of adaptive composition mechanisms $\\mathcal M_t$ for $t\\in [T]$, where\n$\\mathcal M_t$ is $ G_{\\mu_t}$-DP, and each mechanism  works only on a  subsampled data sets by independent Bernoulli trial with probability $p$.  The trade-off function for $\\lim _{T \\rightarrow \\infty} \\otimes_{t=1}^T\n\\left(p\\cdot  G_{\\mu_t}+(1-p) \\text { Id }\\right)\n$ in (\\ref{T-Tradeoff}) approaches to $G_{\\mu_{\\text{tot}}}$-DP when $p\\sqrt{T}$ is a constant, where   \n\\begin{equation}\n\\label{g2dp}\n  \\mu_{\\text{tot}}\n  = p\\cdot \\sqrt{\n  \\sum_{t=1}^{T}\n \\left(\\mathrm{e}^{\\mu_t^2}-1\\right)}.\n\\end{equation}\n\\end{theorem}\nNotably, when $\\mu_t=\\mu$ is substituted for $t\\in [T]$, Eqn.(\\ref{g2dp}) reduces to the CLT result in (\\ref{CLT1}).\n\nTheorem~\\ref{MU-CLT} reflects that, \ngiven the target privacy budget $(\\epsilon,\\delta)$ and the corresponding privacy parameter $ \\mu_{\\text{tot}}$  obtained by (\\ref{compute_mu}),\nwe can proactively allocate privacy cost to each step $t\\in [T]$ for a predefined total number of steps $T$, according to (\\ref{g2dp}). Because $\\mu_t$ is determined by both the clipping value and the noise power, we can adjust both $C_t$ and $\\sigma_t$ to control the privacy budget allocation.\nSections~4.2-4.4 detail the corresponding algorithms.\n\n\n  \n\\subsection{Growing-$\\mu_t$ Method}\nIt is expected that the gradient should be decreased during training, thereby, it is natural to reduce the noise power to stabilize the gradient updates. As a result, we investigate the noise power decay rate in order to satisfy the $\\mu$ grow rate and thus control the privacy cost.\nWith (\\ref{g2dp}) in mind, we  control the privacy cost rate by adjusting a hyper-parameter $\\rho_{\\mu}$ with\n\\begin{equation}\n     \\rho_{\\mu}\\triangleq\\frac{\\mu_{T}}{\\mu_{0}},\n    \\label{c_decay}\\quad \\rho_\\mu\\geq 1.\n\\end{equation}\nThen the dynamic $\\mu_t$, , which corresponds to the dynamic DP-SGD is given by\n\\begin{equation}\n\\label{mut}\n\\mu_t = (\\rho_{\\mu})^{t\/T}\\cdot \\mu_0, \\quad \\forall t\\in [T].\n\\end{equation}\n\n\n\nGiven the total privacy budget $(\\epsilon, \\delta)$, the equivalent  privacy parameter   $\\mu_{\\text{tot}}$ of $\\mu_{\\text{tot}}$-GDP is obtained according to (\\ref{compute_mu}).\nThen the rest problem is to determine the initial state $\\mu_0$. Once  $\\mu_0$ is obtained, the whole $\\{\\mu_t\\}$ sequence can be generated according to (\\ref{mut}).\nBy substituting (\\ref{mut}) into  (\\ref{g2dp}), we obtain \n\\begin{equation}\n\\label{mu0}\n  \\mu^2_{\\text{tot}}\n  = p^2\\cdot  \n  \\sum_{t=1}^{T}\n\\left(\\mathrm{exp}\\left\\{\\left((\\rho_{\\mu})^{t\/T}\\cdot \\mu_0 \\right)^2\\right\\}-1\\right).\n\\end{equation}\n\n\n\\begin{algorithm}[H]\n\n\\caption{$\\mu_t$ computation }\n\\label{alg:mu_allocation}\n\\begin{algorithmic}[1]\n\\REQUIRE    privacy budget $ (\\epsilon, \\delta)$,  $T$, $\\rho_{\\mu}$.\n\\STATE Compute $ \\mu_{\\text{tot}}$ corresponding to $(\\epsilon, \\delta)$ according to (\\ref{compute_mu}).\n\\STATE Compute $\\mu_0$ in  (\\ref{mu0})   by binary search.\n\\STATE Compute $\\sigma_t $ according to (\\ref{noise_power_decay}) for all $t\\in [T]$.\n\\end{algorithmic}\n\\end{algorithm}\n\nBecause the above equation is transcendental when $\\rho_{\\mu}>1$, there is no closed-form solution for $\\mu_0$. Nonetheless, the r.h.s of (\\ref{mu0}) is monotone increasing w.r.t. $\\mu_0$, and we can thus solve it efficiently using a numerical method such as binary search.\nThe computation of $\\mu_t$ is summarized in Algorithm~\\ref{alg:mu_allocation}. The noise power at iteration $t$ can be calculated using a specific expression of $\\mu_0$ and $\\rho_\\mu$:\n\\begin{equation}\n    \\sigma_t = \\frac{C}{\\mu_0}(\\rho_\\mu)^{-\\frac{t}{T}}, \\quad \\rho_{\\mu}>1.\n    \\label{noise_power_decay}\n\\end{equation}\n\n\n\nOne example is shown in Fig.~\\ref{fig:eps_consumption}. Given the total DP budget  $\\epsilon=1.2$, growing-$\\mu_t$ gives the freedom to adjust the privacy cost rate,  which is the slope of the curve. \nIn Fig~\\ref{fig:eps_consumption}, we demonstrate the privacy budget consumption curve for different $\\rho_\\mu$. \nThe solid line ($\\rho_\\mu = 1$) represents vanilla DP-SGD with evenly distributed noise power along with the step updates. With growing-$\\mu_t$, we can now realize any $\\mu$ consumption process under the constraint of the total DP budget as shown by the dashed lines. Specifically, the growing-$\\mu_t$ slows consumption in the early rounds and accelerates consumption in the later rounds.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width = 0.4\\textwidth]{eps2.jpg}\n    \\caption{Consumption of privacy budget with different $\\mu_t$ increasing rates.  } \n    \\label{fig:eps_consumption}\n\\end{figure}\n\n\n\n\n\n\n\n\\vspace{10pt}\n\\subsection{Sensitivity-Decay Method}\nThe gradient norm of a neural network converges to zero once the SGD training converges. This is not the case for DP-SGD.  When the required privacy protection level is high, it requires a large calibrated noise promotional to the clipping value. The updated gradient norm tends to rise, as shown in Fig.~\\ref{fig:Gradient_Norm}. Based on this discovery, we propose to adjust the sensitivity across training iterations.\nWe \nset the evolution of clipping values by\n\\begin{equation}\n   \n    C_t =   (\\rho_{c})^{-\\frac{t}{T}}\\cdot C_0, \\quad \\rho_c\\geq 1.\n    \\label{sens_decay}\n\\end{equation}\nAssuming a constant $\\mu_t$, and by substituting $\\mu_t=\\mu_0$ into (\\ref{g2dp}) in Theorem~1,  we have the closed form solution:\n\\begin{equation}\n    \\mu_0 = \\sqrt{\\log\\left(\\frac{\\mu_{\\text{tot}}^2}{p^2T}+1\\right)}, \\quad t\\in[T].\n    \\label{mut_compute}\n\\end{equation}\nWith (\\ref{sens_decay}) and (\\ref{mut_compute}), we can calibrate the noise power at each round by:\n\\begin{equation}\n     \\sigma_t = \\frac{C_0}{\\mu_0}(\\rho_c)^{-\\frac{t}{T}}, \\quad \\rho_c>1, \\quad t\\in[T].\n    \\label{sigmat_compute_sens_decay}\n\\end{equation}\n \n\n\n\\subsection{Dynamic DP}\n    \\begin{algorithm}[H]\n\\caption{Dynamic DP-SGD Algorithm}\n\\label{alg:NADP}\n\\begin{algorithmic}[1]\n\\REQUIRE  DP budget $(\\epsilon,\\delta)$, sampling rate $p$ and hyper-parameters:  $\\rho_{\\mu}$, $\\rho_{c}$ and $C_0$.\n\\STATE Compute \n$\\mu_0 $ in Algorithm~\\ref{alg:mu_allocation} \n\\FOR{$t=1, \\ldots, T$}\n\\STATE Compute $C_t =   (\\rho_{c})^{-\\frac{t}{T}}\\cdot C_0$ in (\\ref{sens_decay})\n\\STATE Calibrate noise :\n$\\sigma_t = \\frac{C_0}{\\mu_0}(\\rho_{\\mu}\\cdot\\rho_c)^{-\\frac{t}{T}}$\n\\STATE \nSample $ X_t\\in X$ with sampling rate $p$ and sample noise $\\xi_{t}\\sim \\mathcal N(0,\\sigma^2_{t}I)$. \n\\STATE Compute:\n    $\\theta_t =  \\theta_{t-1} - \n    \\frac{\\eta }{\\left|X_t\\right|}\\big[\\xi_{t}+$\n    $ \n    \\sum_{x\\in X_{ t}}\\text{CL}\\left(g_x;C_t\\right)\n    \\big]$ with $\\text{CL}(\\cdot)$ in (\\ref{clipping}) \n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\nBecause the noise calibration is based on the clipping value, we incorporate the growing-$\\mu_t$ method into the sensitivity-decay method and refer to this new one as dynamic DP-SGD.\nIt maintains the same $\\mu_t$ increasing rate as the previous growing-$\\mu_t$ method while having a faster noise decay rate than the sensitivity-decay method.\nWe summarize the dynamic DP-SGD algorithm in Algorithm~\\ref{alg:NADP} and conduct extensive experiments to show how dynamic DP-SGD improves performance.\n\n\n\n\n\\section{Experiments}\n\\subsection{Datasets, Models and Benchmarks}\n\\textbf{Datasets}: To conduct a comprehensive test of the dynamic DP-SGD performance, we run experiments on the following 5 datasets: MNIST, FashionMNIST, IMDB, NAME, and InfiniteMNIST, using neural network models such as MLP, CNN, LSTM, and Federated Learning. In  Appendix~\\ref{apped:dataset},  we describe each data set, the corresponding neural network model, and parameter settings for each experiment separately. We also go into details about the dynamic DP-SGD federated learning algorithm to make the paper  self-contained.\n\n\\textbf{Benchmarks}: Using the aforementioned datasets and models, we compare our proposed dynamic DP-SGD, growing-$\\mu_t$ method, and sensitivity-decay method to the four benchmarks listed below.\n \n\\noindent{(i) \\it{SGD without DP}}: The SGD method serve as the upper bound of model inference accuracy in the absence of clipping and additive noise. \nPlease note that  the DP-SGD and proposed dynamic DP-SGD can also be used to obtain the DP-Adam counterparts of an Adam optimizer due to DP's postprocessing properties~\\citep{dwork2009differential}. For the IMDB data set, we apply the Adam to compute the performance upper bound, and DP-Adam counterparts for comparison.\n\n\\noindent{(ii) \\it{DP-SGD with the CLT of GDP Acountant}~\\citep{bu2019deep}}: GDP, a recently developed DP accounting framework, provides a simple, explicit, and tight privacy accounting CLT bound for the classical DP-SGD with evenly calibrated noise power. It is proved by~\\cite{dong2019gaussian} that the CLT of GDP provides a tighter composition bound for DP-SGD than the moment accountant method~\\citep{DBLP:conf\/ccs\/AbadiCGMMT016}.  As a result, it serves as a standard for our dynamic DP-SGD methods.\n\n\n\\noindent{(iii) {{\\it Noise Power Decay with $\\rho$-zCDP Accountant}} ~\\citep{yu2019differentially}}: This paper proposes decaying the noise power during the SGD and computing the privacy loss using the $\\rho$-zCDP. However, only parallel composition is considered, with no regard for DP amplification by subsampling.\n\n\n\n\\noindent{(iv) \\it{Noise Power Decay with tCDP Accountant}~\\citep{CDP}}:\n\\cite{CDP} recently proposed decaying the noise power for DP-SGD training as well as analyzing the DP composition and subsampling amplification under the truncated concentrated differential privacy (tCDP) framework.\n\n\\textbf{Parameters}: \nWe concentrate on the strong privacy guarantee, with $\\epsilon$ set to be in the range $[0.4, 9]$ and $\\delta = 1\/(10|X|)$, where $|X|$ is the training data sample size. It is worth noting that we observed a significant performance degradation of DP-SGD when performing IMDB tasks. As a result, we also test large $\\epsilon$ values in this case, i.e., $\\epsilon=9$.\nThe hyperparameters $\\rho_c$ and $\\rho_{\\mu}$ are swept in the following predefined sets:\n$\n    1\/\\rho_{c}, 1\/\\rho_{\\mu} \\in  \\{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8\\}.\n   \n   \n   \n$\nOther hyper-parameters are detailed in Appendix~\\ref{apped:dataset}\\footnote{The code will be available at github.com\/dynamic-dp once the paper is accepted.}.\n\n\n\\subsection{Results Analysis}\nExperiment results for the above five different datasets on MLP, CNN, LSTM, and Federated Learning models are shown in Table~1-Table~4, along with benchmarks and proposed methods' performance.\nIt is worth noting that in the majority of cases, we concentrate on the strong privacy protection region with $\\epsilon<1$.\nAs a result, if the DP accountant  is not tight enough, it will lead to an overestimation of the noise power needed, resulting in the noise dominating the gradient and causing the network to fail to learn. For example, the method in~\\cite{yu2019differentially} fails to leverage the subsampling DP amplification in the DP accounting, resulting in significantly worse performance in the high privacy guarantee region due to overestimated noise. As a result, we only replicate its CNN model results in the MNIST and FashionMNIST datasets.\n\nBy contrast, the GDP framework proposed by~\\citet{bu2019deep} provides a detailed examination of DP amplification through subsampling as well as DP composition. Thereby, more precise noise power is calibrated for the same privacy budget. Specifically, even when there are no dynamics for DP-SGD updates, it outperforms the accuracy of the noise decay method by~\\cite{yu2019differentially} and \\cite{CDP}.\n\nThe proposed extended CLT supports the privacy accounting for dynamic clipping and noise power decay. \nSeparate experiments are carried out with the proposed growing-$\\mu_t$ method, sensitivity-decay method, and dynamic DP-SGD method. The results show that all the three proposed methods improve performance when compared to the static noise GDP method~\\cite{bu2019deep}.\nIn particular, \n$\\mu_t$ grows at the expense of early convergence speed in order to achieve higher accuracy, while sensitivity decay ensures more stable convergence. This explains why the performance of sensitivity decay outperforms that of growing-$\\mu_t$. The dynamic DP-SGD, a combination of the two, improves performance while causing no additional privacy loss, as demonstrated by the Dynamic DP results in each table.\n\nExperiments with different privacy budgets were conducted, and the results show that the stronger the privacy protection required (lower $\\epsilon$ value), the more noticeable the improvement by the proposed dynamic DP-SGD method. For example, when $\\epsilon = 0.4$ for the MNIST, even though the noise decay is adopted by~\\cite{yu2019differentially} and~\\cite{zhang2021adaptive}, however, their model fail to learn due to large calibrated noise power by the loose DP compositions. By contrast, our method\noutperforms GDP method by a large margin, achieving $3.17 \\%$ when $\\epsilon = 0.4$ and even $4.6\\%$ for the LSTM network on NAME. Without compromising privacy, our method consistently outperforms all other benchmarks on MNIST, FashionMNIST, IMDN, and NAME datasets for CNN, MLP, LSTM, and federated learning models, as shown in Table1-4.\n\n\n\n\n\\begin{table}[htb!]\n\\centering\n\\caption{CNN on  MNIST and FashionMNIST datasets.}\n\\begin{adjustbox}{width=1\\textwidth}\n\\hfill{}\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline \n\\multirow{2}{*}{DP Accountant}&\\multirow{2}{*}{Dynamic Noise} & \\multicolumn{3}{c|}{ MNIST} & \\multicolumn{3}{c|}{{FashionMNIST}} \n\\tabularnewline\n\\cline{3-5} \\cline{6-8}\n && {$\\epsilon$ = 0.4} & {$\\epsilon$ =0.6}& {$\\epsilon$ =1.2} & {$\\epsilon$ =0.4} & {$\\epsilon$ =1.2}& {$\\epsilon$ =2.0}\\tabularnewline\n\\hline \n\\hline \n{Non-private} &{-}& \\multicolumn{3}{c|}{{98.83}}  & \\multicolumn{3}{c|}{{87.92}}   \\tabularnewline\n\\hline \n\\hline \n{$\\rho$-zCDP \\citep{yu2019differentially}} &Noise Decay& {10.28} & {10.12} & {65.33} & {9.86} & 63.30 & 72.18\n\\tabularnewline\n\\hline \n{tCDP \\citep{CDP}} &Noise Decay& {26.93} & {83.28} & {92.60} & {53.69} & 76.48 & 77.58\n\\tabularnewline\n\\hline\n{CLT for GDP~\\citep{bu2019deep}} &-& {91.18} & {93.80} & {95.50} & {76.77} & 80.45 & 82.55 \\tabularnewline\n\\hline \n\\hline \n\\multirow{3}{*}{Extended CLT for GDP (Ours)}\n&growing-$\\mu_t$  & {91.67} & {94.49} & 96.06& {77.81} & 80.95  & 83.10  \\tabularnewline\n\\cline{2-8}\n & Sensitivity Decay & {93.95} & 95.17 & 96.17 & {78.11} & 82.83  & 83.64  \\tabularnewline\n\n\\cline{2-8}\n&Dynamic DP & {\\textbf{94.35}} & {\\textbf{95.21}} & {\\textbf{96.34}} & {\\textbf{78.50}} & {\\textbf{83.22}} & \\textbf{83.81} \\tabularnewline\n\\hline \n\n\\end{tabular}\\hfill{}\n\\end{adjustbox}\n\\label{mnist_res}\n\\end{table}\n\n\\begin{table}[htb!]\n\\centering\n\\caption{MLP on the IMDB dataset.}\n\\begin{adjustbox}{width=0.9\\textwidth}\n\\hfill{}\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline \n\\multirow{2}{*}{DP Framework}&\\multirow{2}{*}{Dynamic Noise} & \\multicolumn{5}{c|}{ IMDB} \n\\tabularnewline\n\\cline{3-7} \n && {$\\epsilon$ = 0.5} & {$\\epsilon$ =1}& {$\\epsilon$ =3} & {$\\epsilon$ =6} & {$\\epsilon$ =9}\\tabularnewline\n\\hline \n\\hline \n{Non-private} &{-}& \\multicolumn{5}{c|}{{82.85}}  \\tabularnewline\n\\hline \n\\hline\n\n{tCDP \\citep{CDP}} &Noise Decay&56.67 &{58.24} & {62.15} & {65.88} & {70.16} \n\\tabularnewline\n\\hline \n{CLT for GDP \\citep{bu2019deep}} &-& {63.62} & 69.71 & {75.64} & {77.75} & 78.60  \\tabularnewline\n\\hline \n\\hline \n\\multirow{3}{*}{Extended CLT for GDP(Ours)}\n&growing-$\\mu_t$ & {64.92} & {69.85} & 76.00& {78.16} & 78.56  \\tabularnewline\n\\cline{2-7}\n & Sensitivity Decay & {65.44} & {70.25} & 76.23 & {78.47} & 79.42    \\tabularnewline\n\n\\cline{2-7}\n&Dynamic GDP & {\\textbf{65.63}} & {\\textbf{70.77}} & {\\textbf{76.64}} & {\\textbf{78.61}} & {\\textbf{79.61}}  \\tabularnewline\n\\hline \n\n\\end{tabular}\\hfill{}\n\\end{adjustbox}\n\\label{imdb_res}\n\\end{table}\n\n\\begin{table}[htb!]\n\\centering\n\\caption{LSTM on NAME dataset.}\n\\begin{adjustbox}{width=0.8\\textwidth}\n\\hfill{}\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline \n\\multirow{2}{*}{DP Framework}&\\multirow{2}{*}{Dynamic Noise} & \\multicolumn{4}{c|}{ NAME} \n\\tabularnewline\n\\cline{3-6} \n && {$\\epsilon$ = 1} & {$\\epsilon$ =2}& {$\\epsilon$ =4} & {$\\epsilon$ =8} \\tabularnewline\n\\hline \n\\hline \n{Non-private} &{-}& \\multicolumn{4}{c|}{{80.14}}  \\tabularnewline\n\\hline \n\\hline\n\n{tCDP\\citep{CDP}} &Noise Decay&48.13 &{51.20} & {57.67} & {68.53}\n\\tabularnewline\n\\hline \n\n{CLT for GDP~\\citep{bu2019deep}} &-& {62.71} & {69.64} & {73.04} & {74.15}   \\tabularnewline\n\\hline \n\\hline \n\\multirow{3}{*}{Extended CLT for GDP(Ours)}\n&growing-$\\mu_t$ & {64.10} & {71.25} & 74.68& {75.50}   \\tabularnewline\n\\cline{2-6}\n & Sensitivity Decay & {66.66} & {71.78} & {73.67} & {74.78}     \\tabularnewline\n\n\\cline{2-6}\n&Dynamic DP & {\\textbf{67.30}} & {\\textbf{72.01}} & {\\textbf{75.03}} & {\\textbf{75.75}}   \\tabularnewline\n\\hline \n\n\\end{tabular}\\hfill{}\n\\end{adjustbox}\n\\label{name_res}\n\\end{table}\n\n\\begin{table}[htb!]\n\\centering\n\\caption{Federated learning on InfiniteMNIST dataset.}\n\\begin{adjustbox}{width=1\\textwidth}\n\\hfill{}\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline \n\\multirow{2}{*}{DP Framework}&\\multirow{2}{*}{Dynamic Noise} & \\multicolumn{3}{c|}{MNIST-250K} & \\multicolumn{3}{c|}{{MNIST-500K}} \n\\tabularnewline\n\\cline{3-5} \\cline{6-8}\n && {$\\epsilon$ = 0.1} & {$\\epsilon$ =0.4}& {$\\epsilon$ =1} & {$\\epsilon$ =0.1} & {$\\epsilon$ =0.4}& {$\\epsilon$ = 1}\\tabularnewline\n\\hline \n\\hline \n{Non-private} &{-}& \\multicolumn{3}{c|}{{98.89}}  & \\multicolumn{3}{c|}{{98.96}}   \\tabularnewline\n\\hline \n{CLT for GDP~\\citep{bu2019deep}} &-& {93.47} & {95.71} & {96.02} & {94.82} & {96.55} & {96.89} \\tabularnewline\n\\hline \n\\hline \n\\multirow{3}{*}{Extended CLT for GDP(Ours)}\n&growing-$\\mu_t$ & {93.75} & {95.93} & 96.13& {95.65} & {96.76} & 97.05  \\tabularnewline\n\\cline{2-8}\n & Sensitivity Decay & {94.46} & 95.90 & 96.40 & {95.75} & 96.83 & 97.06 \\tabularnewline\n\n\\cline{2-8}\n&Dynamic DP & \\textbf{94.72} & {\\textbf{96.00}} & {\\textbf{96.55}} & \\textbf{95.88} & {\\textbf{96.95}} & {\\textbf{97.22}} \\tabularnewline\n\\hline \n\n\\end{tabular}\\hfill{}\n\\end{adjustbox}\n\\label{FL_res}\n\\end{table}\n\n\\subsection{Hyper-parameter sensitivity}\nWe then test the robustness of dynamic DP-SGD performance to different values of $\\rho_{\\mu}$ and $\\rho_c$ as shown in Fig.~\\ref{fig:decay_rate_search}.\nWe use grid search to demonstrate the impact of these parameters on model performance. The dynamic method can consistently improve model performance across a wide range.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.65\\textwidth]{acc_decay_ratio2.png}\n    \\caption{Dynamic DP-SGD performance is robust to  $\\rho_{\\mu}$ and $\\rho_c$.}\n    \\label{fig:decay_rate_search}\n\\end{figure}\n\\subsection{Exact Privacy Cost}\nThough~\\cite{dong2019gaussian} have shown  that the CLT of GDP approximate the true privacy cost with negligible error,\n\\citet{gopi2021numerical} recently discovers that the CLT of GDP may underestimate the privacy cost.\nThe RDP accountant~\\citep{wang2019subsampled}, on the other hand, overestimates the true cost.\nTo evaluate the exact privacy cost, thereby, we plot the privacy cost curves for both the proposed extended CLT of GDP and RDP in Fig.~\\ref{fig:rdp_gdp_comp}. Specifically, we set a target training round and conduct privacy accounting for the dynamic DP-SGD  based on the extended CLT of GDP as well as RDP\\footnote{For RDP, we use autodp library by \\cite{autodp} for DP accounting.} with the advanced translation method between RDP and $(\\epsilon, \\delta)$-DP~\\citep{balle2020privacy}.\nConsequently,  the true privacy cost curve must lie somewhere between these two limits. The result of extended CLT for GDP is reasonable because the privacy cost differences between these two limits are small, particularly in the high privacy protection region.\n \n\nIt is worth noting that, in addition to the proposed extended CLT for dynamic DP-SGD accounting, exact accounting can be performed using the recently proposed methods by~\\cite{gopi2021numerical} and~\\cite{zhu2021optimal}, separately. However, because they both require numerical computation, given a total privacy budget, the computation of different noise power for  the mechanism of each step  becomes much involved.\n \n\n\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width = 0.35\\textwidth]{comp_rdp_gdp_dynamic_eps_0.1_2.jpg}\n   \n    \\caption{Upper bound (advanced RDP) and lower bound (extended GDP CLT) of the dynamic DP-SGD privacy budget cost curves. The true privacy cost curve must lie somewhere between these two limits. The result of extended CLT for GDP is reasonable because the privacy cost differences between these two limits are small, especially  in the high privacy protection region.}\n    \\label{fig:rdp_gdp_comp}\n\\end{figure}\n\n\n \n\n \n\\section{Conclusions}\nIn this paper,\nwe extend the central limit theorem (CLT) of Gaussian DP to perform tight privacy accounting in order to calibrate dynamic noise for each individual step of stochastic gradient descent (SGD) updates.  \nWe, therefore, are able to allocate a lower privacy cost than the DP-SGD during updates based on this extended CLT until both methods consume the same target privacy budget at the predefined update number.\nExtensive testing on a variety of datasets and models demonstrates that the dynamic DP-SGD consistently and clearly outperforms existing methods.\n\nFor the future work, we intend to investigate how to leverage the  recently achievement of numerical composition methods \\citep{gopi2021numerical, zhu2021optimal} to calibrate the noise power and clipping value for each individual training step of dynamic DP-SGD.\n\n\n\n\\subsubsection*{Acknowledgments}\nWe would like to thank Yuxiang Wang for his helpful discussion.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}