Strong ratio limit theorems associated with random walks

TL;DR

This paper extends strong ratio limit theorems for spread out random walks on unimodular groups to cases where the convergence parameter R is greater than or equal to 1, highlighting the importance of unimodularity.

Contribution

It generalizes previous results to arbitrary R ≥ 1 and clarifies the significance of the unimodular condition for these theorems.

Findings

Extended strong ratio limit theorems to R ≥ 1

Clarified the role of unimodularity in the theorems

Provided a broader framework for random walks on groups

Abstract

Strong ratio limit theorems associated with a broad class of spread out random walks on unimodular groups were proved in the preceding paper, where these random walks were assumed to have the convergence parameter $R=1$. In the present paper, we study the case of an arbitrary $R\ge1$ and clarify the role of the condition that the group is unimodular.