Mathematical Research Letters

Volume 25 (2018)

Number 4

Limit laws for random matrix products

Pages: 1205 – 1212

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n4.a7

Authors

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

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

Abstract

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.

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Received 25 October 2017

Published 16 November 2018