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Frequently visited sets for random walks

Endre Csáki·Antónia Földes·Pál Révész·Jay Rosen·Zhan Shi
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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 $\sup_{x\in Z^d}μ_n^X(x+A)/\log n$ tends to a finite limit as $n\to\infty$. 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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Endre Csáki, Antónia Földes, Pál Révész, Jay Rosen, Zhan Shi. 2004-12-01. Frequently visited sets for random walks. https://arxiv.org/abs/math/0412018

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