Conditioned random walks on linear groups II: local limit theorems

TL;DR

This paper proves a local limit theorem for conditioned random walks on linear groups, analyzing their asymptotic behavior and exit times, by combining advanced probabilistic techniques for forward and reversed walks.

Contribution

It establishes the local limit theorem for conditioned random walks on linear groups, extending previous work with new methods for the reversed walk analysis.

Findings

Proved the local limit theorem for conditioned random walks on linear groups.

Derived the local behavior of exit times for these walks.

Combined Caravenna-type theorem with conditioned CLT for reversed walks.

Abstract

We investigate random walks on the general linear group constrained within a specific domain, with a focus on their asymptotic behavior. In a previous work [38], we constructed the associated harmonic measure, a key element in formulating the local limit theorem for conditioned random walks on groups. The primary aim of this paper is to prove this theorem. The main challenge arises from studying the conditioned reverse walk, whose increments, in the context of random walks on groups, depend on the entire future. To achieve our goal, we combine a Caravenna-type conditioned local limit theorem with the conditioned version of the central limit theorem for the reversed walk. The resulting local limit theorem is then applied to derive the local behavior of the exit time.