On Products of Nonnegative Matrices

Cohn, Harry ; Nerman, Olle

Ann. Probab., Tome 18 (1990) no. 4, p. 1806-1815 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1990-10-14
Classification:  Nonnegative matrix,  stochastic matrix,  Markov chain,  space-time chain,  Harmonic function,  tail $\sigma$-field,  atomic set,  ratio limit,  weak ergodicity,  15A48,  60F99,  60J05,  60J45