Limit laws for random matrix products

TL;DR

This paper investigates the asymptotic behavior of products of matrices with a normalization, showing convergence to the matrix exponential under certain conditions, and illustrates the results with a hyperbolic geometry example.

Contribution

It establishes a limit law for normalized products of matrices and provides a dynamical perspective and an example involving hyperbolic geometry.

Findings

Normalized matrix products converge to the exponential of the mean matrix.

The result applies to sequences with bounded mean norms.

An example demonstrates the application to hyperbolic random walks.

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, the product $\left(I\_d + \frac1n A\_0 \right) \dots \left(I\_d + \frac1n A\_{n-1} \right) $ 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.