Frequently visited sets for random walks

We study the occupation measure of various sets for a symmetric transient random walk in Zd with finite variances. Let μnX(A) denote the occupation time of the set A up to time n. It is shown that supx∈ZdμnX(x+A)/logn tends to a finite limit as n→∞. The limit is expressed in terms of the largest eigenvalue of a matrix involving the Green function of X restricted to the set A. Some examples are discussed and the connection to similar results for Brownian motion is given.