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

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