Limit laws for random matrix products
Résumé
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.
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