Jetlir Duraj, Kilian Raschel, Pierre Tarrago, Vitali Wachtel
We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.
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The Martin boundary associated with the simple random walk on an example of partially oriented lattice is shown to be trivial by computing fine estimates of the Green kernel.
We investigate Martin boundary for a non-centered random walk on ${\mathbb Z}^d$ killed up on the time $τ_\vartheta$ of the first exit from a convex cone with a vertex at $0$. The approach combines large deviation estimates, the ratio limite theorem and the ladder height process. The results are applied to identify the Martin boundary for a random walk killed upon the first exit from a convex cone having $C^1$ boundary.
We derive diffusion constants and martingales for senile random walks with the help of a time-change. We provide direct computations of the diffusion constants for the time-changed walks. Alternatively, the values of these constants can be derived from martingales associated with the time-changed walks. Using an inverse time-change, the diffusion constants for senile random walks are then obtained via these martingales. When the walks are diffusive, weak convergence to Brownian motion can be shown using a martingale functional limit theorem.
In this paper, martingales related to simple random walks and their maximum process are investigated. First, a sufficient condition under which a function with three arguments, time, the random walk, and its maximum process becomes a martingale is presented, and as an application, an alternative way of deriving the Kennedy martingale is provided. Then, a complete characterization of a function with two arguments, the random walk and its maximum, being a martingale is presented. This martingale can be regarded as a discrete version of the Azéma--Yor martingale. As applications of discrete Azéma…
We give a power law for the homogeneity degree of the Martin kernel of the fractional Laplacian for the right circular cone when the angle of the cone tends to zero.
The t-Martin boundary of a random walk on a half-space with reflected boundary conditions is identified. It is shown in particular that the t-Martin boundary of such a random walk is not stable in the following sense : for different values of t, the t-Martin compactifications are not homeomorphic to each other.
We give a simple non-analytic proof of Biggins' theorem on martingale convergence for branching random walks.
We consider controlled random walks that are martingales with uniformly bounded increments and nontrivial jump probabilities and show that such walks can be constructed so that P(S_n^u=0) decays at polynomial rate n^{-α} where α>0 can be arbitrarily small. We also show, by means of a general delocalization lemma for martingales, which is of independent interest, that slower than polynomial decay is not possible.
In this note, we study the Martin boundary of the simple random walk on an example of directed graph. More precisely, the Martin boundary is shown to be trivial, i.e. there is no non-constant positive harmonic functions. The proof involves fine estimates of the Green function which are summarized in this note.
We extend the notion of the associated random walk and the Wald martingale in random walks where the increments are independent and identically distributed to the more general case of stationary ergodic increments. Examples are given where the increments are Markovian or Gaussian, and an application in queueing is considered.
We compute the $t$-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete $t$-harmonic functions. Our approach is uniform in $t$, and shows that there are three regimes for the Martin boundary.
Suppose that $(X,Y,Z)$ is a random walk in $\mathbb{Z}^3$ that moves in the following way: on the first visit to a vertex only $Z$ changes by $\pm 1$ equally likely, while on later visits to the same vertex $(X,Y)$ performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.
A complete representation of the Martin boundary of killed random walks on a half-space $\Z^{d-1}\times\N^*$ is obtained. In particular, it is proved that the corresponding Martin boundary is homemorphic to the half-sphere ${\cal S}^d_+ = \{z\in\R^{d-1}\times\R_+ : |z|=1\}$. The method is based on a combination of ratio limits theorems and large deviation techniques.
Martin boundaries of diffusions and random walks on hyperbolic spaces
本文研究双曲空间上特定扩散过程及其在等距群上关联随机游走的Martin边界。
In a seminal paper Biggins and Kyprianou \cite{BKy04} proved the existence of a non degenerate limit for the {\it Derivative martingale} of the branching random walk. As shown in \cite{Aid11} and \cite{Mad11}, this is an object of central importance in the study of the extremes of the branching random walk. In this paper we investigate the tail distribution of the limit of the {\it Derivative Martingale} by mean of the study of the global minimum of the branching random walk. This new approach leads us to extend the results of \cite{Gui90} and \cite{Bur09} under slighter assumptions.
Consider a real-valued branching random walk in the boundary case. Using the techniques developed by Aïdékon and Shi [5], we give two integral tests which describe respectively the lower limits for the minimal position and the upper limits for the associated additive martingale.
Biggins [Uniform convergence of martingales in the branching random walk. {\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $λ$ from an open set $Λ\subseteq \mathbb{C}^d$. We investigate the martingales corresponding to parameters from the boundary $\partial Λ$ of $Λ$. The boundary can be decomposed into several parts. There may be a part of the boundary, on which the martingales do not exist, on other parts it exists, but diverges or vanishes in the limit. In the rema…
$1$-stable fluctuation of the derivative martingale of branching random walk
本文研究了分支随机游走导数鞅波动的函数律收敛，得到了比Buraczewski等人更强的结果，并建立了与分支布朗运动情形相对应的结论。
The Martin compactification is investigated for a d-dimensional random walk which is killed when at least one of it's coordinates becomes zero or negative. The limits of the Martin kernel are represented in terms of the harmonic functions of the associated induced Markov chains. It is shown that any sequence of points x_n with lim_n |x_n| = \infty and lim_n x_n/|x_n|= q is fundamental in the Martin compactification if up to the multiplication by constants, the induced Markov chain corresponding to the direction q has a unique positive harmonic function. The full Martin compactification is obta…
Exponential moments of truncated branching random walk martingales
本文研究漂移至无穷远的分支随机游走的马尔萨斯鞅，证明原点以下粒子被清除时该鞅极限呈指数尾（无清除时为多项式尾），临界情形下截断导数鞅也有相同结论，该研究受Erdős–Rényi图首次通过渗流相关工作启发。
We consider a branching random walk on the line. Biggins and Kyprianou [6] proved that, in the boundary case, the associated derivative martingale converges almost surly to a finite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In the present paper, we give a necessary and sufficient condition for the non-triviality of this limit and establish a Kesten-Stigum-like result.
Let $W_n, n\in\mn_{0}$ be an intrinsic martingale with almost sure limit $W$ in a supercritical branching random walk. We provide criteria for the $L_p$-convergence of the series $\sum_{n\ge 0} e^{an}(W-W_n)$ for $p>1$ and $a>0$. The result may be viewed as a statement about the exponential rate of convergence of $\me |W-W_n|^p$ to zero.
We compute the exponential decay of the probability that a given multi-dimensional random walk stays in a convex cone up to time $n$, as $n$ goes to infinity. We show that the latter equals the minimum, on the dual cone, of the Laplace transform of the random walk increments. As an example, our results find applications in the counting of walks in orthants, a classical domain in enumerative combinatorics.
We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp.\ non-rational), then the Martin boundary is countable, homeomorphic to $\mathbb Z\cup\{\pm\infty\}$ (resp.\ uncountable, homeomorphic to $\mathbb R\cup\{\pm\infty\}$). Such instability phenomena are very rare in the literature. Along the way of proving th…
Martin boundaries and asymptotic behavior of branching random walks
该文研究无限局部有限图上超临界暂留分支随机游走与Martin边界的关系，发现t-Martin边界与标准Martin边界的新联系，还分析了子图上分支游走的存活与持续特性并给出相关示例。
Let $(W_n(θ))_{n\in\mathbb N_0}$ be the Biggins martingale associated with a supercritical branching random walk and denote by $W_\infty(θ)$ its limit. Assuming essentially that the martingale $(W_n(2θ))_{n\in\mathbb N_0}$ is uniformly integrable and that $\text{Var} W_1(θ)$ is finite, we prove a functional central limit theorem for the tail process $(W_\infty(θ) - W_{n+r}(θ))_{r\in\mathbb N_0}$ and a law of the iterated logarithm for $W_\infty(θ)-W_n(θ)$, as $n\to\infty$.
Branching random walk conditioned on large martingale limit
本文研究非边界情形分支随机游走，以鞅极限或最小值为条件，建立联合尾部分布并证明从最小值视角的分布收敛，兼得导数鞅极限右尾。
Let $(Y_n)$ be a sequence of i.i.d. real valued random variables. Reflected random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1} = |X_n - Y_{n+1}|$. In this note, we study recurrence of this process, extending a previous criterion. This is obtained by determining an invariant measure of the embedded process of reflections.
The complete representation of the Martin compactification for reflected random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial'' compactification obtained by Ney and Spitzer for the homogeneous random walks in $\Z^d$ : convergence of a sequence of points $z_n\in\Z^{d-1}\times\N$ to a point of on the Martin boundary does not imply convergence of the sequence $z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large deviation properties of the scaled processes and uses Pascal's method…
We consider a branching random walk on $\mathbb{R}$ with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$ and $\tilde Z_n(t)=\int e^{tx}Z_n(dx)$ be its Laplace transform. We show the $L^p$ convergence rate and the uniform convergence of the martingale $\tilde Z_n(t)/\mathbb E[\tilde Z_n(t)|ξ]$, and establish a moderate deviation principle for the measures $Z_n$.
A complete representation of the Martin boundary of killed random walks on the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is obtained. It is proved that the corresponding full Martin compactification of the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is homeomorphic to the closure of the set $\{w={z}/{(1+|z|)}:z\in{\mathbb{N}}^*\times{\mathbb{N}}^*\}$ in ${\mathbb{R}}^2$. The method is based on a ratio limit theorem for local processes and large deviation techniques.
Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is known since the work of Ancona and Gou{ë}zel-Lalley that the Martin boundary coincides with the geometric boundary. The goal of this paper is to weaken the finite support assumption. We first show that, in any non-amenable group, there exist probability measures with exponential tails giving rise to pathological Martin boundaries. Then, for probability measur…
Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment
本文研究了在时间非齐次随机环境中的分支随机游走导数鞅收敛性，通过Tanaka分解确定了其非平凡性条件。
We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. The case of Galton-Watson processes is particularly included so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering. As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be interesting on its own and which correct, generalize and simplify some earlier works.
Consider the random walk $S_n=ξ_1+...+ξ_n$ with independent and identically distributed increments and negative mean $\mathbf Eξ=-m<0$. Let $M=\sup_{0\le i} S_i$ be the supremum of the random walk. In this note we present derivation of asymptotics for $\mathbf P(M>x), x\to\infty$ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory of long-tailed distributions. In addition the same approach allows to obtain asymptotics for $\mathbf P(M_τ>x)$, where $M_τ=\max_{0\le i<τ}S_i$ and $τ=\min\{n\ge 1: S_n\le 0 \}$.
Spinal decomposition, martingale convergence and the Seneta-Heyde scaling for matrix branching random walks
本文研究了非负矩阵半群上的矩阵分支随机游走，推导了加性鞅的 Biggins 收敛定理类似结果、脊柱分解定理、导数鞅收敛及 Seneta-Heyde 标度，并提供了中心化 Markov 随机游走更新测度的显式对偶结果。
Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection, first observed by the second author in [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.]…
The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walks, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for…
This paper is devoted to the asymptotic analysis of the reinforced elephant random walk (RERW) using a martingale approach. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the RERW. The distributional convergences of the RERW to some Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the RERW. All our analysis relies on asymptotic results for multi-dimensional martingales with matrix normalization.
Scaling limit of boundary random walks: A martingale problem approach
本研究采用鞅问题方法结合弱收敛技术，证明一类边界随机游走在适当标度下于J₁-Skorokhod拓扑中收敛于半直线上的一般布朗运动，其核心创新是边界随机游走局部时的渐近行为结果，可推导相关布朗型极限过程的CLT结果。
We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [9]. Our method applies furthermore to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [17]. Surprisingly, this phase transition phenomenon disappears in the sense …
The left tail of the subcritical derivative martingale in a branching random walk
本文在高斯分支随机游走框架下，精确估计了$L^4$相中导数鞅分布的左尾，回答了Lacoin等人提出的猜想。
We work under the A\"ıdékon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by $Z$. It is shown that $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+o(\log x)$ as $x\to\infty$. Also, we provide necessary and sufficient conditions under which $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+{\rm const}+o(1)$ as $x\to\infty$. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the …
We consider killed planar random walks on isoradial graphs. Contrary to the lattice case, isoradial graphs are not translation invariant, do not admit any group structure and are spatially non-homogeneous. Despite these crucial differences, we compute the asymptotics of the Martin kernel, deduce the Martin boundary and show that it is minimal. Similar results on the grid $\mathbb Z^d$ are derived in a celebrated work of Ney and Spitzer.
We consider the random walks killed at the boundary of the quarter plane, with homogeneous non-zero jump probabilities to the eight nearest neighbors and drift zero in the interior, and which admit a positive harmonic polynomial of degree three. For these processes, we find the asymptotic of the Green functions along all infinite paths of states, and from this we deduce that the Martin compactification is the one-point compactification.
The purpose of this paper is to establish, via a martingale approach, some refinements on the asymptotic behavior of the one-dimensional elephant random walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory parameter $p$ which lies between zero and one. This behavior is totally different in the diffusive regime $0 \leq p <3/4$, the critical regime $p=3/4$, and the superdiffusive regime $3/4<p \leq 1$. Notwithstanding of this trichotomy, we provide some new results on the almost sure convergence and the asymptotic normality of the ERW.
In this paper we define a general setting for Martin boundary theory associated to quantum random walks, and prove a general representation theorem. We show that in the dual of a simply connected Lie subgroup of U(n), the extremal Martin boundary is homeomorphic to a sphere. Then, we investigate restriction of quantum random walks to Abelian subalgebras of group algebras, and establish a Ney-Spitzer theorem for an elementary random walk on the fusion algebra of SU(n), generalizing a previous result of Biane. We also consider the restriction of a quantum random walk on $SU_q(n)$ introduced by I…
Enumeration of three quadrant walks with small steps and walks on other M-quadrant cones
该研究将四分之一平面小步游走计数的相关结果推广到任意M象限锥情形，推导了刻画生成函数的解析泛函方程，明确了不同M值下生成函数的代数、D有限或D代数性质。
In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation.…
Martingale Approach to Gambler's Ruin Problem for Correlated Random Walks
本文利用鞅的可选停时定理研究相关随机游走的赌徒破产问题，推导了特定增量和对称带延迟情况下的破产概率与期望时长的闭式表达式，并开发了针对一般带延迟CRW的鞅技术及对任意模式进行下注的破产概率研究。
In this paper we give a simple new proof of a result of Pittel and Wormald concerning the asymptotic value and (suitably rescaled) limiting distribution of the number of vertices in the giant component of $G(n,p)$ above the scaling window of the phase transition. Nachmias and Peres used martingale arguments to study Karp's exploration process, obtaining a simple proof of a weak form of this result. We use slightly different martingale arguments to obtain a much sharper result with little extra work.
We prove invariance principles for a mulditimensional random walk conditioned to stay in a cone. Our first result concerns convergence towards the Brownian meander in the cone. Furthermore, we prove functional convergence of $h$-transformed random walk to the corresponding $h$-transform of the Brownian motion. Finally, we prove an invariance principle for bridges of a random walk in a cone.
We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel-Leader graph $DL(q,r)$, where $q,r \ge 2$. The latter is the horocyclic product of two homogeneous trees with respective degrees $q+1$ and $r+1$. When $q=r$, it is the Cayley graph of the wreath product (lamplighter group) ${\mathbb Z}_q \wr {\mathbb Z}$ with respect to a natural set of generators. We describe the full Martin compactification of these random walks on $DL$-graphs and, in particular, lamplighter groups. This completes and provides a better approach to prev…
Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green function of the corresponding random walk. It is known from the work of W. Woess that when a finitely supported random walk on a free product of abelian groups is adapted to the free product structure, the Martin boundary coincides with the geometric boundary. The main goal of this paper is to deal with non-adapted finitely supported random walks, for which there is no explicit formula for the Green function. Nevertheless, we show that the Martin boundary still coinci…
Space-time boundaries for random walks and their application to operator algebras
本文研究了随机游走的空间-时间马丁边界，并将其与算子代数中的紧致化相关联，证明了0-马丁边界与λ-马丁核的联系，揭示了非交换谢洛夫边界与Toeplitz C*代数的一致性。
We consider multidimensional discrete valued random walks with nonzero drift killed when leaving general cones of the euclidian space. We find the asymptotics for the exit time from the cone and study weak convergence of the process conditioned on not leaving the cone. We get quasistationarity of its limiting distribution. Finally we construct a version of the random walk conditioned to never leave the cone.
Let $\mm_n, n=0,1,...$ be the supercritical branching random walk, in which the number of direct descendants of one individual may be infinite with positive probability. Assume that the standard martingale $W_n$ related to $\mm_n$ is regular, and $W$ is a limit random variable. Let $a(x)$ be a nonnegative function which regularly varies at infinity, with exponent greater than -1. The paper presents sufficient conditions of the almost sure convergence of the series $\sum_{n=1}^{\infty}a(n)(W-W_n)$. Also we establish a criterion of finiteness of $\me W\log^+ W a(\log^+W)$ and $\me \log^+|\zi| a(…
This paper studies the asymptotic behavior of the Green function of a multidimensional random walk killed when leaving a convex cone with smooth boundary. Our results imply uniqueness, up to a multiplicative factor, of the positive harmonic function for the killed random walk.
Martin boundary of a space-time Brownian motion with drift killed at the boundary of a moving cone
本文研究了在运动锥边界消亡的带漂移时空布朗运动，利用解析方法和递归补偿方法确定了其抛物线Martin边界、调和函数、逃逸概率及转移核显式公式。
A local limit theorem for nonlattice multidimensional random walks in cones
研究非格点随机游走在一般锥体中的渐进行为，结合随机游走在锥体中的积分定理和经典无限制随机游走定理，证明局部极限定理。
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