BERRY–ESSEEN BOUND AND LOCAL LIMIT THEOREM FOR THE COEFFICIENTS OF PRODUCTS OF RANDOM MATRICES

Abstract Let $\mu $ be a probability measure on $\mathrm {GL}_d(\mathbb {R})$ , and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law . Under assumptions that has finite exponential moment generates proximal strongly irreducible semigroup, we prove Berry–Esseen bound optimal rate $O(1/\sqrt n)$ for coefficients of $S_n$ settling long-standing question considered since fundamental work Guivarc’h Raugi. The local limit theorem is also obtained, complementing recent partial result Grama, Quint Xiao.