Strong law of large numbers for a function of the local times of a transient random walk in ${\mathbb{Z}}^d$

Abstract

For an arbitrary transient random walk $(S_n{)}_{n\ge 0}$ in ${\mathbb{Z}}^d$, $d\ge 1$, we prove a strong law of large numbers for the spatial sum ${\sum }_{x\in {\mathbb{Z}}^d}f(l(n,x))$ of a function $f$ of the local times $l(n,x)={\sum }_{i=0}^n\mathbb{I}\{S_i=x\}$. Particular cases are the number of (a) visited sites (first time considered by Dvoretzky and Erdős), which corresponds to a function $f(i)=\mathbb{I}\{i\ge 1\}$; (b) $\alpha$-fold self-intersections of the random walk (studied by Becker and König), which corresponds to $f(i)=i^{\alpha }$; (c) sites visited by the random walk exactly $j$ times (considered by Erdős and Taylor and by Pitt), where $f(i)=\mathbb{I}\{i=j\}$.