Consider $(g_n)_{n\geq 1}$ a sequence of independent and identically distributed random matrices and the left random walk $G_n : = g_n \ldots g_1,$ $n\geq 1$ on the general linear group $GL(d, \mathbb R)$.

In this talk, I will present a Bahadur-Rao-Petrov type large deviation expansion for the coefficient $\langle f, G_n v \rangle$ of the product $G_n$, where $v \in \mathbb R^d$ and $f \in (\mathbb R^d)^*$.

A local limit theorem with large deviations for the coefficients will also be presented.

The talk is based on a joint work with Ion Grama (Univ. Bretagne-Sud) and Hui Xiao (Chinese Academy of Sciences), to appear in Ann. Prob.