Moments and Distribution of the Local Times of a Transient Random Walk on ℤd

Consider an arbitrary transient random walk on ℤd with d∈ℕ. Pick α∈[0,∞), and let Ln(α) be the spatial sum of the αth power of the n-step local times of the walk. Hence, Ln(0) is the range, Ln(1)=n+1, and for integers α, Ln(α) is the number of the α-fold self-intersections of the walk. We prove a strong law of large numbers for Ln(α) as n→∞. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by Černý (Stoch. Proc. Appl. 117:262–270, 2007). Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.