Moments and distribution of the local times of a transient random walk on $\Z^d$

Abstract

Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $\alpha\in[0,\infty)$ and let $L_n(\alpha)$ be the spatial sum of the $\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $\alpha$, $L_n(\alpha)$ is the number of the $\alpha$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(\alpha)$ as $n\to\infty$. 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 \v{C}ern\'y \cite{Ce07}. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.