In an important but little known paper published in 1937 (Le cas discontinu des
probabilites en chaine, Pub. Fac. Sci. Univ. Masaryk (Brno) 236) Doeblin
introduced a number of finite nonhomogeneous Markov chain models and gave
without proofs several results concerning their asymptotic behaviour. As was the
case with
many of his contributions to Probability, Doeblin's insight into the mechanism of
the process was well ahead of
his time. It is shown that using the tail $\sfield$ of the chain as well as
considerations pertaining to Martin boundary theory one can derive general
results for the asymptotics of nonhomogeneous Markov chains. Extensions to
ratio limit theorems for nonnegative nonstochastic matrices are also given.