Non-stationary version of Furstenberg Theorem on random matrix products

TL;DR

This paper extends Furstenberg's theorem to non-stationary settings, showing that under certain conditions, the norms of products of independent, non-identically distributed matrices grow exponentially, with a predictable asymptotic behavior.

Contribution

It introduces a non-stationary version of Furstenberg's theorem, providing conditions for exponential growth and asymptotic description of matrix product norms.

Findings

Norms of matrix products grow exponentially under generic conditions.

Existence of a non-random sequence describing asymptotic behavior.

Extension of Furstenberg's theorem to non-stationary matrix sequences.

Abstract

We prove a non-stationary analog of the Furstenberg Theorem on random matrix products (that can be considered as a matrix version of the law of large numbers). Namely, under a suitable genericity conditions the sequence of norms of random products of independent but not necessarily identically distributed $\SL(d, \mathbb{R})$ matrices grow exponentially fast, and there exists a non-random sequence that almost surely describes asymptotical behaviour of that sequence.