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January 2004 The escape rate of favorite sites of simple random walk and Brownian motion
Mikhail A. Lifshits, Zhan Shi
Ann. Probab. 32(1A): 129-152 (January 2004). DOI: 10.1214/aop/1078415831

Abstract

Consider a simple symmetric random walk on the integer lattice $\ZB$. For each n, let $V(n)$ denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417--436] says that V is almost surely transient, thus disproving a previous conjecture of Erdős and Révész [Mathematical Structures--Computational Mathematics--Mathematical Modeling 2 (1984) 152--157]. More precisely, Bass and Griffin proved that almost surely, $\liminf_{n\to \infty} {|V(n)| \over n^{1/2}(\log n)^{-\gamma}}$ equals $0$ if $\gamma<:1$, and is infinity if $\gamma>11$ (eleven). The present paper studies the rate of escape of $V(n)$. We show that almost surely, the "lim\,inf'' expression in question is 0 if $\gamma\leq 1$, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.

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Mikhail A. Lifshits. Zhan Shi. "The escape rate of favorite sites of simple random walk and Brownian motion." Ann. Probab. 32 (1A) 129 - 152, January 2004. https://doi.org/10.1214/aop/1078415831

Information

Published: January 2004
First available in Project Euclid: 4 March 2004

zbMATH: 1067.60072
MathSciNet: MR2040778
Digital Object Identifier: 10.1214/aop/1078415831

Subjects:
Primary: 60G50, 60J55, 60J65

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 1A • January 2004
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