  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 + \frac1n A\_0 \right) \dots
\left(I\_d + \frac1n 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.
