Mathematical Research Letters

Volume 25 (2018)

Number 4

Limit laws for random matrix products

Pages: 1205 – 1212



Jordan Emme (Laboratoire de Mathémathiques, Université Paris-Sud, CNRS Université Paris-Saclay, Orsay, France)

Pascal Hubert (Aix-Marseille Université, CNRS Centrale Marseille, France)


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, we prove that the product $(I_d + \frac{1}{n} A_0) \dotsc (I_d + \frac{1}{n} A_{n-1})$ 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.

Received 25 October 2017

Accepted 15 January 2018

Published 16 November 2018