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Register a new userHidden symmetries and limit laws in the extreme order statistics of the Laplace random walk
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This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching representation and a squared Bessel representation. These complementary descriptions expose various hidden symmetries in branching processes and Brownian motion which lie behind some striking formulas found by Schehr and Majumdar (Phys. Rev. Lett., 108:040601). In particular, the Bessel process of dimension $4 = 2+2$ appears in the descriptions as a path decomposition of Brownian motion at a local minimum and the Ray-Knight description of Brownian local times near the minimum.
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This paper is concerned with the limit theory of the extreme order statistics derived from random walks. We establish the joint convergence of the order statistics near the minimum of a random walk in terms of the Feller chains. Detailed descriptions of the limit process are given in the case of simple symmetric walks and Gaussian walks. Some open problems are also presented.
A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_n)_{nin mathbb{N}cup{0}}$ in a sparse random environment $(S_k,lambda_k)_{kinmathbb{Z}}$ is a nearest neighbor random walk on $mathbb{Z}$ that jumps to the left or to the right with probability $1/2$ from every point of $mathbb{Z}setminus {ldots,S_{-1},S_0=0,S_1,ldots}$ and jumps to the right (left) with the random probability $lambda_{k+1}$ ($1-lambda_{k+1}$) from the point $S_k$, $kinmathbb{Z}$. Assuming that $(S_k-S_{k-1},lambda_k)_{kinmathbb{Z}}$ are independent copies of a random vector $(xi,lambda)in mathbb{N}times (0,1)$ and the mean $mathbb{E}xi$ is finite (moderate sparsity) we obtain stable limit laws for $X_n$, properly normalized and centered, as $ntoinfty$. While the case $xileq M$ a.s. for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $mathbb{E} xi=infty$ (strong sparsity) will be analyzed in a forthcoming paper.
In this article, we consider a Branching Random Walk (BRW) on the real line where the underlying genealogical structure is given through a supercritical branching process in i.i.d. environment and satisfies Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (depends on the expected size of the $n$-th generation given the environment) maximum among positions at the $n$-th generation converges weakly to a scale-mixture of Frech{e}t random variable. Furthermore, we derive the weak limit of the extremal processes composed of appropriately scaled positions at the $n$-th generation and show that the limit point process is a member of the randomly scaled scale-decorated Poisson point processes (SScDPPP). Hence, an analog of the predictions by Brunet and Derrida (2011) holds.
We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit that is interesting in its own right. We describe the limit object as a growing collection of lilypads built on a Poisson point process in $mathbb{R}^d$. As an application of our main theorem, we show that the maximizer of the system displays the ageing property.
We consider a random walk $tilde S$ which has different increment distributions in positive and negative half-planes. In the upper half-plane the increments are mean-zero i.i.d. with finite variance. In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation. For the distributions with a slowly varying tails, we show that ${frac{1}{sqrt n} tilde S(nt)}$ has no weak limit in $De$; alternatively, the weak limit is a reflected Brownian motion.
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