BERRY-ESSEEN BOUNDS AND MODERATE DEVIATIONS FOR RANDOM WALKS ON GL d (R)

Overview & Synopsis

Let (gn) n 1 be a sequence of independent and identically distributed random elements of the general linear group GL d (R), with law µ. Consider the random walk Gn := gn. .. g1. Denote respectively by Gn and ρ(Gn) the operator norm and the spectral radius of Gn. For log Gn and log ρ(Gn), we prove moderate deviation principles under exponential moment and strong irreducibility conditions on µ; we also establish moderate deviation expansions in the normal range [0, o(n 1/6)] and Berry-Esseen bounds under the additional proximality condition on µ. Similar results are found for the couples (X x n , log Gn) and (X x n , log ρ(Gn)) with target functions, where X x n := Gn • x is a Markov chain and x is a starting point on the projective space P(R d).