Berry-Esseen bounds for products of random matrices under a $2+\delta$ polynomial moment

Abstract

Let $(g_k{)}_{k\ge 1}$ be a sequence of independent and identically distributed matrices with law $\mu$ on $\mathrm{GL}(d,\mathbb{R})$, where $d\ge 2$ is an integer. Suppose that the closed semigroup generated by the support of $\mu$ is strongly irreducible and proximal and that the polynomial moment of order $2+\delta$ is finite, where $0<\delta \le 1$. We establish a Berry-Esseen bound of order $n^{-\delta /2}$ for $\log \Vert g_n\cdots g_{1}v\Vert$, where $v$ is a nonzero vector in ${\mathbb{R}}^d$ and $\Vert \cdot \Vert$ is an arbitrary norm on ${\mathbb{R}}^d$. As an application, we also obtain a Berry-Esseen bound of order $n^{-\delta /2}$ for $\log \Vert g_n\cdots g_1\Vert$, where $\Vert \cdot \Vert$ is any matrix norm. The proof combines an $m$-dependent approximation with Stein's method. The argument uses identities for exchangeable pairs constructed by resampling individual matrices, together with a concentration inequality for sums of the approximating variables.