A representation for products of finite non--negative matrices is given in
terms of products of stochastic matrices and as a result Markov chains
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.