Precise large deviation asymptotics for products of random matrices

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 $\mathrm{\mid }{G}_{n}x\mathrm{\mid }$, where $G_n:=g_n\dots g_1$. The goal of this paper is to establish precise asymptotics for large deviation probabilities $\mathbb P(\log | G_n x | \geq n(q+l))$,where$q>\gamma $isfixedand$l$isvanishingas$n\to \infty$. We study both invertible matrices and positive matrices and give analogous results for the couple $(X_n^x,\log \mathrm{\mid }{G}_{n}x\mathrm{\mid })$ with target functions, where $X_n^x=G_{n}x\mathrm{/}\mathrm{\mid }{G}_{n}x\mathrm{\mid }$. As applications we improve previous results on the large deviation principle for the matrix norm $\mathrm{\parallel }{G}_n\mathrm{\parallel }$ and obtain a precise local limit theorem with large deviations.