Products of stochastic matrices: Exact rate for convergence in probability for directed networks

We study the products W_k···W₁ of random stochastic, not necessarily symmetric matrices. It is known that, under certain conditions, the product W_k · · · W₁ converges almost surely (a.s.) to a random rank-one matrix; the latter is equivalent to |λ₂(W_k · · · W₁)| → 0 a.s., where λ₂(·) is the second largest (in modulus) eigenvalue. In this paper, we show that the probability that |λ₂(W_k · · · W₁)| stays above ε ∈ (0,1] in the long run decays to zero exponentially fast ~ e^-kI. Furthermore, we explicitly characterize the rate of this convergence I and show that it depends only on the underlying graphs that support the matrices W_k´s. Our results reveal that the rate I is essentially determined by the most likely way in which the union (over time) of the support graphs fails to form a directed tree.