On the behavior of random walk around heavy points
type: Item
instance of: scholarly article
title: On the behavior of random walk around heavy points
zbMATH Open document ID: 1151.60021
author: Endre Csáki
author: Antónia Földes
author: Pál Révész
published in: Journal of Theoretical Probability
publication date: 2008-02-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0605221
review text: The random walk \(S_{0}=0,S_n=\sum_{k=1}^n X_k,n=1,2,\dots \) on the d-dimensional integer lattice \(\mathbb{Z}^d\) is considered where \(X_k\), \(k=1,2,\dots \) are i.i.d. random variables with the distribution \(P\{ X_{1}=x\} =p(x) ,x\in \mathbb{Z}^d\) such that \(\sum_{x\in \mathbb{Z}^d}| x|^{2}p(x) dx<\infty ,d=3\), \(\sum_{x\in \mathbb{Z}^d}| x|^{2}\log (| x| +1) p(x) <\infty ,d=4\), \(\sum_{x\in \mathbb{Z}^d}| x|^{d-2}p(x) dx<\infty ,d\geq5\). The covariance matrix of \(X\) is \(Q\). \(| Q| \) is its determinant and \(Q^{-1}\) is its inverse. The random walk \(\{ S_n\} \) is supposed to be aperiodic. Its Green function \(G( x) :=\sum_{n=0}^\infty P\{Sn=x\} ,x\in \mathbb{Z}^d\) has the asymptotic property \(G( x) \sim C_d| Q|^{-\frac{1}{2} }\| x\|^{2-d}\) as \(| x| \to \infty ,\| x\|^{2}=xQ^{-1}x\), with a certain constant \(C_d\). Denote \(\gamma =\frac{1}{G( 0)};1-\gamma_{x}=\frac{G( x) }{G( 0) },x\neq 0\). The local time is as follows \(\xi ( x,n) :=\sum_{k=1}^n I\{ S_k=x\} ,n=1,2,\dots ,x\in \mathbb{Z}^d\). The random walk is transient for \(d\geq 3\), i.e. any point can have only a finite number of visits, but there are (random) points that accumulate a higher number of visits, the so-called heavy points. The problem is to find out wheather there can be another heavy point or a point not visited at all up to time \(n\) in the close neighborhood of a heavy point. The main statements in this paper imply the negative answer. Denote \(\xi ( n) :=\max_{x\in \mathbb{Z}^d}\xi ( x,n) ,n=1,2,\dots \), \(\eta ( n) :=\max_{1\leq j\leq n}\xi ( S_{j},\infty ) ,n=1,2,\dots\). Even for the general aperiodic random walk \(\lim_{n\to \infty }\frac{\eta ( n) }{\log n} =\lambda \) a.s. with \(\lambda =-\frac{1}{\log ( 1-\gamma ) }\). Define \[ m_{x}=\begin{cases} 1&\text{if }x<0 \\ \frac{( 1-\gamma _{x}) ^{2}}{1-\gamma}& \text{if }x\neq 0.\end{cases} \] Let \(k_n=( 1-\delta _n) \lambda \log n\), where \(\delta _n>0\) is non-increasing. Let \(r_n>0\) be non-decreasing and for…
reviewed by: Nijolė Kalinauskaitė
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 5236406
zbMATH Keywords: Random walk in d-dimension
zbMATH Keywords: Local time
zbMATH Keywords: Occupation time
zbMATH Keywords: Strong theorems
MaRDI profile type: Publication
OpenAlex ID: W1995490456
arXiv ID: math/0605221
cites work: Heavy points of a \(d\)-dimensional simple random walk
cites work: Frequently visited sets for random walks
cites work: Joint asymptotic behavior of local and occupation times of random walk in higher dimension
cites work: Some problems concerning the structure of random walk paths
cites work: Some intersection properties of random walk paths
cites work: The range of transient random walk
cites work: The maximum of the local time of a transient random walk
cites work: Q5709396
cites work: Q5731217
cites work: Green's Functions for Random Walks on ZN
DOI: 10.1007/S10959-007-0092-Z
description: scientific article