Berry-Esseen bounds with targets and Local Limit Theorems for products of random matrices

Abstract

Let $\mu$ be a probability measure on ${\text{GL}}_d(\mathbb{R})$ and denote by $S_n:=g_n\cdots g_1$ the associated random matrix product, where $g_j$'s are i.i.d.'s with law $\mu$. We study statistical properties of random variables of the form $$\sigma (S_n,x)+u(S_{n}x),$$ where $x\in {\mathbb{P}}^{d-1}$, $\sigma$ is the norm cocycle and $u$ belongs to a class of admissible functions on ${\mathbb{P}}^{d-1}$ with values in $\mathbb{R}\cup \{\pm \infty \}$. Assuming that $\mu$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry-Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on $\mathbb{R}$ and Hölder continuous target functions on ${\mathbb{P}}^{d-1}$. As particular cases, we obtain new limit theorems for $\sigma (S_n,x)$ and for the coefficients of $S_n$.

Major modifications from v1: more general results with unified proofs allowing admissible functions u. Results for the norm cocycle and the coefficients are now particular cases. Proof of the Local Limit Theorem included. arXiv admin note: text overlap with arXiv:2110.09032