Berry-Esseen type bounds for the Left Random Walk on GL d (R) under polynomial moment conditions

TL;DR

This paper establishes Berry-Esseen bounds for the logarithm of the norm of products of random matrices in GL(d,R), under polynomial moment conditions, improving previous convergence rate results.

Contribution

It provides new Berry-Esseen bounds for the left random walk on GL(d,R) under polynomial moment assumptions, with explicit convergence rates.

Findings

Rate of convergence is ((log n)/n)^{q/2-1} for moments of order q in (2,3].

Rate improves to 1/√n when the moment order is 4.

Results significantly enhance previous bounds in this setting.

Abstract

Let $A_n= \varepsilon_n \cdots \varepsilon_1$, where $(\varepsilon_n)_{n \geq 1}$ is a sequence of independent random matrices taking values in $ GL_d(\mathbb R)$, $d \geq 2$, with common distribution $\mu$. In this paper, under standard assumptions on $\mu$ (strong irreducibility and proximality), we prove Berry-Esseen type theorems for $\log ( \Vert A_n \Vert)$ when $\mu$ has a polynomial moment. More precisely, we get the rate $((\log n) / n)^{q/2-1}$ when $\mu$ has a moment of order $q \in ]2,3]$ and the rate $1/ \sqrt{n} $ when $\mu$ has a moment of order $4$, which significantly improves earlier results in this setting.