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

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.