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

TL;DR

This paper establishes optimal Berry-Esseen bounds and Local Limit Theorems for certain random variables derived from products of random matrices, extending classical limit theorems to matrix products with broad observables.

Contribution

It provides new limit theorems and bounds for matrix product-related variables under minimal moment conditions and broad classes of functions, improving previous results in the field.

Findings

Optimal Berry-Esseen bounds for matrix product variables

Local Limit Theorem for observables on matrix products

New limit theorems for coefficients of matrix products

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 H\"older 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 $S_n$.