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# 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

Published 16 November 2018