Asymont, Inna and Korshunov, Dmitry
(2019)
*Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$.*
Journal of Theoretical Probability.
ISSN 0894-9840

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## Abstract

For an arbitrary transient random walk $(S_n)_{n\ge 0}$ in $Z^d$, $d\ge 1$, we prove a strong law of large numbers for the spatial sum $\sum_{x\in Z^d}f(l(n,x))$ of a function $f$ of the local times $l(n,x)=\sum_{i=0}^n I\{S_i=x\}$. Particular cases are the number of (a) visited sites (first time considered by Dvoretzky and Erd\H{o}s), which corresponds to a function $f(i)=I\{i\ge 1\}$; (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 $f(i)=I\{i=j\}$.