Limit laws for random matrix products

摘要

In this short note, we study the behaviour of a product of matrices with a simultaneous renormalization. Namely, for any sequence (A(n))(n)(is an element of N) of d x d complex matrices whose mean A exists and whose norms' means are bounded, we prove that the product (I-d + 1/n A(0)) ... (I-d + 1/n A(n-1)) 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.