Analytic Dependence of the Lyapunov Moment Function and the Projective Stationary Measure for Random Matrix Products

TL;DR

This paper proves the local analyticity of the Lyapunov moment function and stationary measure for i.i.d. random matrix products, leading to insights into the regularity of Lyapunov exponents and moments.

Contribution

It establishes the local analyticity of the Lyapunov moment function and stationary measure for random matrix products, a novel result in the field.

Findings

Lyapunov moment function is locally analytic in the measure space.

Unique stationary measure on projective space is analytically dependent on the measure.

Asymptotic variance and higher-order moments are also analytic.

Abstract

We consider the product of i.i.d. random matrices sampled according to a probability measure $\mu$ supported on a strongly irreducible and proximal subset of a compact set $S\subset GL(d,\mathbb{R})$. We establish the local analyticity of the Lyapunov moment function and the unique stationary measure on the projective space with respect to $\mu$ in the total variation topology. As a consequence, we obtain the analyticity of the asymptotic variance and all higher-order Lyapunov moments.