Hitting times for the perturbed reflecting random walk
type: Item
instance of: scholarly article
title: Hitting times for the perturbed reflecting random walk
zbMATH Open document ID: 1259.82045
author: Laurent Serlet
published in: Stochastic Processes and their Applications
publication date: 2012-11-15T00:00:00Z
review text: For the real valued process \((X_n)_{n\geq 0}\), denote by \(F_n^X\) the \(\sigma\)-algebra generated by \(X_0, X_1, \dots, X_n\) and set \(\overline{X}_n=\max\{X_0, X_1,\dots, X_n\}\). The perturbed reflecting random walk (PRRW) with reinforcement parameter \(r\in (-1,1)\) is a process \((X_n)_{n\geq 0}\) taking its values on \(\mathbb{Z}_+=\{0,1,2,\dots\}\) such that, for every \(n\geq 0\), \(X_{n+1}\in \{X_n-1, X_n+1\}\) and the transition probability \(\mathbb{P}(X_{n+1}=X_n+1 | F_n^X)\) is equal to \(\frac12\) if \(0<X_n<\overline{X}_n\); \(\frac{1-r}2\) if \(X_n={\overline{X}_n}\) and \(n\geq 1\); 1 if \(X_n=0\). Moreover \(X_0=0\). The goal of the paper is to study the PRRW via an excursion point of view. The law of the hitting times is computed. The invariance principles with explicit descriptions of the asymptotic laws are given. Some results on the almost sure asymptotic behavior are obtained.
reviewed by: U. A. Rozikov
Mathematics Subject Classification ID: 82B41
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 6106884
zbMATH Keywords: perturbed random walk
zbMATH Keywords: once reinforced random walk
zbMATH Keywords: perturbed Brownian motion
zbMATH Keywords: hitting times
zbMATH Keywords: invariance principle
zbMATH Keywords: recurrence
zbMATH Keywords: law of the iterated logarithm
MaRDI profile type: Publication
full work available at URL: j.spa.2012.09.003
OpenAlex ID: W1979285033
DOI: 10.1016/J.SPA.2012.09.003
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description: scientific article; zbMATH DE number 6106884