Precise large deviation asymptotics for products of random matrices

Abstract

Let (gn)n⩾1 be a sequence of independent identically distributed d×d real random matrices with Lyapunov exponent λ. For any starting point x on the unit sphere in Rd, we deal with the norm |Gnx|, where Gn≔gn…g1. The goal of this paper is to establish precise asymptotics for large deviation probabilities P(log|Gnx|⩾n(q+l)), where q>λ is fixed and l is vanishing as n→∞. We study both invertible matrices and positive matrices and give analogous results for the couple (Xnx,log|Gnx|) with target functions, where Xnx=Gnx∕|Gnx|. As applications we improve previous results on the large deviation principle for the matrix norm ‖Gn‖ and obtain a precise local limit theorem with large deviations.