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

## TL;DR

This paper establishes a strong law of large numbers for sums involving functions of local times of a transient random walk in integer lattice spaces, unifying and extending previous results on visited sites and self-intersections.

## Contribution

It proves a general strong law of large numbers for functions of local times, covering various specific cases like visited sites, self-intersections, and sites visited exactly j times.

## Key findings

- Proves a strong law of large numbers for sums of functions of local times.
- Unifies previous results on visited sites and self-intersections.
- Extends the understanding of local time functionals in transient random walks.

## 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 $f(i)=\mathbb I\{i=j\}$.