Berry–Esseen type bounds for the left random walk on GLd(R) under polynomial moment conditions

Abstract

Let ${\mathit{A}}_{\mathit{n}}={\mathit{\epsilon }}_{\mathit{n}}\cdots {\mathit{\epsilon }}_1$, where ${({\mathit{\epsilon }}_{\mathit{n}})}_{\mathit{n}\ge 1}$ is a sequence of independent random matrices, taking values in ${\mathrm{GL}}_{\mathit{d}}(\mathbb{R})$, $\mathit{d}\ge 2$, with common distribution μ. In this paper, under standard assumptions on μ (strong irreducibility and proximality) we prove Berry–Esseen type theorems for $log(\Vert {\mathit{A}}_{\mathit{n}}\Vert )$ when μ has a polynomial moment. More precisely, we get the rate ${((log\mathit{n})/\mathit{n})}^{\mathit{q}/2-1}$, when μ has a moment of order $\mathit{q}\in ]2,3]$ and the rate $1/\sqrt{\mathit{n}}$ when μ has a moment of order 4, which significantly improves earlier results in this setting.