Some Limit Theorems Regarding Products of Random Matrices I: Directional Derivative of the Lyapunov Exponent

TL;DR

This paper investigates the asymptotic behavior of products of random matrices, focusing on the directional derivatives of the Lyapunov exponent and their implications for the action on projective space and the unit circle.

Contribution

It introduces new limit theorems for sequences involving products of random matrices and analyzes their impact on the Lyapunov exponent's directional derivatives.

Findings

Derived limit theorems for matrix product sequences.

Analyzed asymptotic behavior on projective space.

Explored action on the unit circle.

Abstract

Given an i.i.d. sequence $\{A_n(\omega)\}_{n\ge 1}$ of invertible matrices and a random matrix $B(\omega)$, we consider the random matrix sequences inductively defined by $S_n(\omega) = A_n(\omega)S_{n-1}(\omega)$ and $T_n(\omega) = B(\sigma^{n-1}\omega)S_{n-1}(\omega)+A_n(\omega)T_{n-1}(\omega)$, and study several limit theorems involving $T_n(\omega)$ as well as the asymptotic behaviour of the action of $T_n(\omega)$ on the projective space and on the unit circle.