Almost Sure Convergence of Products of 2× 2 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 × 2 nonnegative matrices, with finite support A = {A(0), . . . , A(s − 1)}, and assuming that at least one of the A(k) is not diagonal, the normalized columns of the product matrix Pn = A(ω1) . . . A(ωn) 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 PnV ‖PnV ‖ also converges almost surely with an exponential rate of convergence. On the other hand if