Frequently visited sets for random walks

2004/12/01 by Endre Csáki, Antónia Földes, Csáki, Endre +7
Mathematics · #60G50 (primary) 60F15 #60J55 (secondary) #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F15 #msc:60G50 #msc:60J55

paper · pdf · doi:10.48550/arxiv.math/0412018

19 pages

arxiv created 2004/12/01 · arxiv updated 2009/12/01

Abstract

We study the occupation measure of various sets for a symmetric transient random walk in Z^d with finite variances. Let μ^X_n(A) denote the occupation time of the set A up to time n. It is shown that sup_{x∈ Z^d}μ_n^X(x+A)/log n tends to a finite limit as n→∞. The limit is expressed in terms of the largest eigenvalue of a matrix involving the Green's function of X restricted to the set A. Some examples are discussed and the connection to similar results for Brownian motion is given.

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