XXIV.—Conditions for the Ergodicity of Non-homogeneous Finite Markov Chains

In this note we study the asymptotic behaviour of a product of matrices where P j is a matrix of transition probabilities in a non-homogeneous finite Markov chain. We give conditions that (i) the rows of P ( n ) tend to identity and that (ii) P ( n ) tends to a limit matrix with identical rows.