Berry-Esseen Bounds with Targets and Local Limit Theorems for Products of Random Matrices

Abstract

Let mu be a probability measure on GL(d)(R) and denote by S-n := g(n) middot middot middot g(1) the associated random matrix product, where gj'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 is an element of Pd-1, sigma is the norm cocycle and u belongs to a class of admissible functions on P(d-1 )with values in RU{+/-infinity l. 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 R and Holder continuous target functions on Pd-1. As particular cases, we obtain new limit theorems for sigma (S-n,S- x) and for the coefficients of S-n.