Limit laws for random matrix products

Abstract

In this short note, we study the behaviour of a product of matrices with a simultaneous renormalization. Namely, for any sequence $(A\_n)\_n\in \mathbb{N}$ of $d\times d$ complex matrices whose mean $A$ exists and whose norms' means are bounded, the product $\left(I\_d+\frac{1}{n}A\_0\right)\dots \left(I\_d+\frac{1}{n}A\_n-1\right)$ converges towards $\exp A$. We give a dynamical version of this result as well as an illustration with an example of "random walk" on horocycles of the hyperbolic disc.