Exponential growth of products of non-stationary Markov-dependent matrices

TL;DR

This paper establishes conditions under which the product of non-stationary Markov-dependent matrices exhibits exponential growth, extending classical results from i.i.d. and stationary Markov cases to more general non-stationary settings.

Contribution

It generalizes Furstenberg's theorem to non-stationary Markov-dependent matrices, providing new criteria for exponential growth of matrix products.

Findings

Provides sufficient conditions for exponential growth of matrix products.

Extends classical theorems to non-stationary Markov-dependent matrices.

Generalizes previous results from i.i.d. and stationary Markov cases.

Abstract

Let $(\xi_j)_{j\ge1} $, be a non-stationary Markov chain with phase space $X$ and let $\mathfrak{g}_j:\,X\mapsto\mathrm{SL}(m,\mathbb{R})$ be a sequence of functions on $X$ with values in the unimodular group. Set $g_j=\mathfrak{g}_j(\xi_j)$ and denote by $S_n=g_n\ldots g_1$, the product of the matrices $g_j$. We provide sufficient conditions for exponential growth of the norm $\|S_n\|$ when the Markov chain is not supposed to be stationary. This generalizes the classical theorem of Furstenberg on the exponential growth of products of independent identically distributed matrices as well as its extension by Virtser to products of stationary Markov-dependent matrices.