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

Abstract

Let $(g_{k})_{k \geq 1}$ be a sequence of independent and identically distributed matrices with law $μ$ on $\mathrm{GL}(d,\mathbb{R})$, where $d\geq2$ is an integer. Suppose that the closed semigroup generated by the support of $μ$ is strongly irreducible and proximal and that the polynomial moment of order $2+δ$ is finite, where $0<δ\leq 1$. We establish a Berry-Esseen bound of order $n^{-δ/2}$ for $\log \| g_{n} \cdots g_{1} v\|$, where $v$ is a nonzero vector in $\mathbb{R}^{d}$ and $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^{d}$. As an application, we also obtain a Berry-Esseen bound of order $n^{-δ/2}$ for $\log \| g_{n} \cdots g_{1}\|$, where $\|\cdot\|$ 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.