Strong law of large numbers for a function of the local times of a transient random walk on groups

TL;DR

This paper establishes a strong law of large numbers for functions of local times of transient random walks on groups, extending previous results on integer lattices using subadditive ergodic theory.

Contribution

It generalizes the strong law of large numbers for local times to a broader class of groups under weaker conditions, utilizing subadditive ergodic theorem techniques.

Findings

Almost sure convergence of local time functions

Convergence in $L^1$ and $L^2$ for certain functions

Extension of previous lattice results to group settings

Abstract

This paper presents the strong law of large numbers for a function of the local times of a transient random walk on groups, extending the research of Asymont and Korshunov for random walks on the integer lattice $\mathbb{Z}^d$. Under some weaker conditions, we prove that certain function of the local times converges almost surely and in $L^1$ and $L^2$. The proof is mainly based on the subadditive ergodic theorem.