Irina Ignatiouk-Robert
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.
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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.
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…
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…
Martin boundary of a degenerate Reflected Brownian Motion in a wedge
本文研究退化反射布朗运动在楔形中的马丁边界，通过计算格林函数的拉普拉斯变换并推导其渐进行为，确定了最小马丁边界及相应的调和函数。
Martingales On A Euclidean Manifold With A Boundary And Reflected BSDES In Non-Convex Domains
本文研究了带有边界的欧几里得流形上的Γ-martingale与非凸域中反射BSDEs的解的存在性和唯一性，提出了一种基于随机几何的新方法。
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 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 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.
Let $π$ be the occupancy density of an obliquely reflected Brownian motion in the half plane and let ($ρ$, $α$) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of $π$($ρ$, $α$) as $ρ$ $\rightarrow$ $\infty$ with $α$ $\in$ (0, $π$). We find explicit functions a, b, c such that $π$($ρ$, $α$) $\sim$ $ρ$$\rightarrow$$\infty$ a($α$)$ρ$ b($α$) e --c($α$)$ρ$. This closes an open problem first stated by Professor J. Michael Harrison in August 2013. We also compute the exact asymptotics for the tail distribution of the boundary occupancy m…
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 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 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.
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…
The joint distribution of value and local time for Brownian Motion has been reported by Borodin and Salminen. Its asymptotic behavior for recurrent random walk has been presented by Jain and Pruitt. Motivated by the need for queue size control during a pandemic (Hassin, Meilijson and Perlman), the current study presents closed form formulas for random walk and reflected random walk with $\pm 1$ increments, not necessarily fair.
$1$-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.
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。
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…
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 …
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 …
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 this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Kni…
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.…
Conditions for Equivalence of Random Interlacements and Random Walk Reflected off of Infinity
研究证明随机交错与随机游走反射于无穷远在特定条件下等价，条件为所有调和函数为常数或生成森林一致。
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(…
Let $(Y_n)$ be a sequence of i.i.d. $\mathbb Z$-valued random variables with law $μ$. The reflected random walk $(X_n)$ is defined recursively by $X_0=x \in \mathbb N_0, X_{n+1}=|X_n+Y_{n+1}|$. Under mild hypotheses on the law $μ$, it is proved that, for any $ y \in \mathbb N_0$, as $n \to +\infty$, one gets $\mathbb P_x[X_n=y]\sim C_{x, y} R^{-n} n^{-3/2}$ when $\sum_{k\in \mathbb Z} kμ(k) >0$ and $\mathbb P_x[X_n=y]\sim C_{y} n^{-1/2}$ when $\sum_{k\in \mathbb Z} kμ(k) =0$, for some constants $R, C_{x, y}$ and $C_y >0$.
We define the reflection of a random walk at a general barrier and derive, in case the increments are light tailed and have negative mean, a necessary and sufficient criterion for the global maximum of the reflected process to be finite a.s. If it is finite a.s., we show that the tail of the distribution of the global maximum decays exponentially fast and derive the precise rate of decay. Finally, we discuss an example from structural biology that motivated the interest in the reflection at a general barrier.
Boundary contacts for reflected random walks in the quarter plane
本文研究四分之一平面反射随机游走在边界轴上所花时间的极限分布，通过耦合方法和补偿方法分别在不同假设下揭示其递归结构并给出显式闭式表达式。
Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity
本文定义了在瞬态无限图上从无穷远反射的随机游走，证明了Aldous-Broder算法和Wilson算法可生成自由均匀生成森林，并应用于超临界刘维尔量子引力中的随机平面地图。
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.
Reflected random walk in higher dimension arises from an ordinary random walk (sum of i.i.d. random variables): whenever one of the reflecting coordinates becomes negative, its sign is changed, and the process continues from that modified position. One-dimensional reflected random walk is quite well understood from work in 7 decades, but the multidimensional model presents several new difficulties. Here we investigate recurrence questions.
Sample-path large deviations for unbounded additive functionals of the reflected random walk
本文研究 Lindley 递归 Markov 链无界泛函的样本路径大偏差，利用再生周期分解在 M_1' 拓扑下证明次线性速度 LDP，并揭示 MRW 繁忙期面积的重尾特性。
We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an $α$-stable law with $1<α<2$. We prove that the derivative martingale $D_n$ converges to a non-trivial limit $D_\infty$ under some regular conditions. We also study the additive martingale $W_n$, and prove $n^\frac{1}αW_n$ converges in probability to a constant multiple of $D_\infty$.
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.
We consider branching random walks with a spine in the domain of attraction of an $α$-stable Lévy process. For this process, the classical derivative martingale in general degenerates in the limit. We first determine the quantity replacing the derivative martingale and show that it converges to a non-degenerate limit under a certain LlogL-type condition which we assume to be optimal. We go on to give the Seneta-Heyde norming for the critical additive martingale under the same assumptions. The proofs are based on the methods introduced in our previous paper which considered the finite variance …
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.
In this work, we focus on the stationary analysis of a specific class of continuous time Markov-modulated reflected random walks in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series appr…
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.
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。
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