Limit theorems for inhomogeneous random walks on $GL(d,\mathbb R)$

TL;DR

This paper establishes limit theorems such as Berry-Esseen, invariance principles, and large deviations for inhomogeneous random walks on the general linear group, under certain contraction and boundedness conditions.

Contribution

It extends classical limit theorems to products of non-identically distributed matrices with new techniques involving reversing time and variance analysis.

Findings

Proved Berry-Esseen bounds for matrix products.

Established almost sure invariance principles with explicit rates.

Characterized divergence of the variance of the logarithm of the product norm.

Abstract

We prove Berry-Esseen theorems, almost sure invariance principle rates and large deviations for products of independent but not identically distributed invertible matrices with some average (logarithmic) projective contraction and uniform boundedness assumptions. We also characterize the divergence of the variance of the logarithm of the norm of the product. Our approach is based on verifying the conditions of \cite{NewBE} after reversing time.