Almost sure convergence of products of $2\times2$ nonnegative matrices

TL;DR

This paper investigates the almost sure convergence properties of products of 2x2 nonnegative matrices, establishing conditions under which normalized columns converge exponentially and the rank of limit points.

Contribution

It provides new criteria for exponential convergence and rank-one limit points in products of 2x2 nonnegative matrices under probabilistic assumptions.

Findings

Normalized columns converge almost surely with exponential rate if Lyapunov exponents are distinct.

Limit points of normalized products are rank 1 under certain matrix conditions.

Convergence rates can be exponential or not, depending on matrix structure.

Abstract

We study the almost sure convergence of the normalized columns in an infinite product of nonnegative matrices, and the almost sure rank one property of its limit points. Given a probability on the set of $2\times2$ nonnegative matrices, with finite support $\mathcal A=\{A(0),\dots,A(s-1)\}$, and assuming that at least one of the $A(k)$ is not diagonal, the normalized columns of the product matrix $P_n=A(\omega_1)\dots A(\omega_n)$ converge almost surely (for the product probability) with an exponential rate of convergence if and only if the Lyapunov exponents are almost surely distinct. If this condition is satisfied, given a nonnegative column vector $V$ the column vector $\frac{P_nV}{\Vert P_nV\Vert}$ also converges almost surely with an exponential rate of convergence. On the other hand if we assume only that at least one of the $A(k)$ do not have the form $\begin{pmatrix}a&0\\0&d\end{pmatrix}$, $ad\ne0$, nor the form $\begin{pmatrix}0&b\\d&0\end{pmatrix}$, $bc\ne0$, the limit-points of the normalized product matrix $\frac{P_n}{\Vert P_n\Vert}$ have almost surely rank 1 -although the limits of the normalized columns can be distinct- and $\frac{P_nV}{\Vert P_nV\Vert}$ converges almost surely with a rate of convergence that can be exponential or not exponential.