Contents Online
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
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.
Received 25 October 2017
Accepted 15 January 2018
Published 16 November 2018