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

TL;DR

The paper proves a strong law of large numbers for a function of the local times of a transient random walk on a countable group, extending results known for random walks on $\mathbb{Z}^d$. Under a tail condition on $f$, it shows $\frac{1}{n}G_n(f)$ converges to $\gamma^2\sum_{j\ge1} f(j)(1-\gamma)^{j-1}$ both almost surely and in $L^1$, with a further $L^2$ convergence result under a stronger, square-summability-type condition $\sum_{j\ge1} f(j)^2(1-\gamma)^j/j<\infty$. The main technique is the subadditive ergodic theorem, complemented by detailed analysis of the range structure via $R_n^{(k)}$ and truncation arguments to manage tails. The work also discusses quasi-optimality: the $L^2$ condition is essentially tight, as certain natural choices of $f$ fail to yield $L^2$ convergence, highlighting the balance between tail decay and convergence behavior.

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.

Theorems & Definitions (12)

• Theorem 1
• Remark 1
• Remark 2
• Theorem 2
• Remark 3
• Lemma 1: Subadditive Ergodic Theorem
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['thm: main']}
• Lemma 3