Limit laws for random matrix products

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$complexmatriceswhosemean$A$ exists and whose norms’ means are bounded, the product $\left(I_d + \frac1n A_0 \right) \dots \left(I_d + \frac1n A_{n-1} \right) $convergestowards$\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.