Frequently visited sets for random walks

Abstract

We study the occupation measure of various sets for a symmetric transient random walk in $Z^d$ with finite variances. Let ${\mu }_n^X(A)$ denote the occupation time of the set $A$ up to time $n$. It is shown that ${\sup }_{x\in Z^d}{\mu }_n^X(x+A)/\log n$ 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.

19 pages