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

Abstract

Let mu be a probability measure on GLd(R), and denote by Sn := gn middot middot middotg1 the associated random matrix product, where gj 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/root n) for the coefficients of Sn, settling a long-standing question considered since the fundamental work of Guivarc'h and Raugi. The local limit theorem for the coefficients is also obtained, complementing a recent partial result of Grama, Quint and Xiao.