Frequently visited sets for random walks

We study the occupation measure of various sets for a symmetric transient random walk in Z d with finite variances. Let μ n X ( 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 function of X restricted to the set A. Some examples are discussed and the connection to similar results for Brownian motion is given.