Strong law of large numbers for a function of the local times of a transient random walk in $\mathbb Z^d$ (1908.06611v1)

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 Erd\H{o}s), 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 K\"{o}nig), which corresponds to $f(i)=i^\alpha$; (c) sites visited by the random walk exactly $j$ times (considered by Erd\H{o}s and Taylor and by Pitt), where $f(i)=\mathbb I{i=j}$.