Laws of the iterated logarithm for the range of random walks in two and three dimensions
type: Item
instance of: scholarly article
title: Laws of the iterated logarithm for the range of random walks in two and three dimensions
zbMATH Open document ID: 1031.60031
author: Richard F. Bass
author: Takashi Kumagai
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Let \((S_n)\) be a random walk in \(Z^d\), i.e. the sum of i.i.d. centered random variables \(X_i\) taking values in \(Z^d\). Let \(R_n\) be the range of the random walk, i.e. \(R_n\) is the number of distinct sites visited by \(S_0, S_1, \dots ,S_n\). For \(d=3\), under an additional assumption on the moments of \(X_1\), the authors prove an almost sure invariance principle for \(R_n\). As a corollary, they obtain laws of the iterated logarithm. For \(d=2\), assuming that the two coordinates of \(X_1\) are independent and bounded, they prove a law of the iterated logarithm for the range \(R_n\).
reviewed by: Nina Gantert
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 1906089
zbMATH Keywords: range of random walk
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: almost sure invariance principle
zbMATH Keywords: intersection local time
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1029867131
OpenAlex ID: W2011604975
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DOI: 10.1214/AOP/1029867131
description: scientific article; zbMATH DE number 1906089