Precise large deviation asymptotics for products of random matrices

Abstract

Let $(g_n{)}_{n\ge 1}$ be a sequence of independent identically distributed $d\times d$ real random matrices with Lyapunov exponent $\gamma$. For any starting point $x$ on the unit sphere in ${\mathbb{R}}^d$, we deal with the norm $|G_{n}x|$, where $G_n:=g_n\dots g_1$. The goal of this paper is to establish precise asymptotics for large deviation probabilities $\mathbb{P}(\log |G_{n}x|\ge n(q+l))$, where $q>\gamma$ is fixed and $l$ is vanishing as $n\to \infty$. We study both invertible matrices and positive matrices and give analogous results for the couple $(X_n^x,\log |G_{n}x|)$ with target functions, where $X_n^x=G_{n}x/|G_{n}x|$. As applications we improve previous results on the large deviation principle for the matrix norm $\Vert G_n\Vert$ and obtain a precise local limit theorem with large deviations.