Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices

Abstract

Consider the random matrix products Gn := gn . . . g1, where (gn) n 1 is a sequence of independent and identically distributed positive random d × d matrices for any integer d 2. For any starting point x ∈ R d + with |x| = 1 and y 0, consider the exit time τx,y = inf{k 1 : y + log |G k x| < 0}. In this paper, we study the conditioned local probability P(y + log |Gnx| ∈ [0, ∆] + z, τx,y > n) under various assumptions on y and z. For y = o(√ n), we prove precise upper and lower bounds when z is in a compact interval and give exact asymptotics when z → ∞. We also study the case when y ≍ √ n and establish the corresponding asymptotics in function of the behaviour of z.