Asymptotics of Products of Nonnegative Random Matrices

Asymptotic properties of products of random matrices $\xi_k=X_k\cdots X_1$ as $k\to\infty$ are analyzed. All product terms $X_i$ are independent and identically distributed on a finite set of nonnegative matrices $\mathcal{A}=\{A_1,\dots, A_m\}$. We prove that if $\mathcal{A}$ is irreducible, then all nonzero entries of the matrix $\xi_k$ almost surely have the same asymptotic growth exponent as $k\to\infty$, which is equal to the largest Lyapunov exponent $\lambda(\mathcal{A})$. This generalizes previously known results on products of nonnegative random matrices. In particular, this removes all additional “nonsparsity” assumptions on matrices imposed in the literature. We also extend this result to reducible families. As a corollary, we prove that Cohen's conjecture (on the asymptotics of the spectral radius of products of random matrices) is true in case of nonnegative matrices.