On Products of Nonnegative Matrices

Abstract

A representation for products of finite nonnegative matrices is given in terms of products of stochastic matrices and as a result Markov chain arguments are used to derive ratio limit properties. In particular, we obtain necessary and sufficient conditions for weak ergodicity and give a probabilistic proof of the Coale-Lopez theorem. In the general case, there are several sequences of sets of partitions of the state space corresponding to an associated nonhomogeneous Markov chain which lead to a number of ratio product limits. Asymptotic column proportionality, characteristic of weak ergodicity, may occur only inside each sequence of sets with one possible exception.