Berry-Esseen bound and Local Limit Theorem for the coefficients of 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 $g_j$ are i.i.d. with law $\mu$. Under the assumptions that $\mu$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry-Esseen bound with the optimal rate $O(1/\sqrt{n})$ and a general Local Limit Theorem for the coefficients of $S_n$.

27 pages. arXiv admin note: text overlap with arXiv:2106.04019

References in corpus (3)

• A zero-one law for invariant measures and a local limit theorem for coefficients of random walks on the general linear group
• Large deviation expansions for the coefficients of random walks on the general linear group
• Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices

Cited by in corpus (1)

• Limit theorems for the coefficients of random walks on the general linear group