Generalized Dobrushin ergodicity coefficient and ergodicities of non-homogeneous Markov chains

In our earlier paper, a generalized Dobrushin ergodicity coefficient of Markov operators (acting on abstract state spaces) with respect to a projection P has been introduced and studied. It turned out that the introduced coefficient was more effective than the usual ergodicity coefficient. In the present work, by means of left consistent Markov projections and the generalized Dobrushin’s ergodicity coefficient, we investigate uniform and weak P-ergodicities of non-homogeneous discrete Markov chains (NDMC) on abstract state spaces. It is easy to show that uniform P-ergodicity implies a weak one, but in general, the reverse is not true. Therefore, some conditions are found which together with weak P-ergodicity of NDMC imply its uniform P-ergodicity. Furthermore, necessary and sufficient conditions are found by means of the Doeblin’s condition for the weak P-ergodicity of NDMC. The weak P-ergodicity is also investigated in terms of perturbations. Several perturbative results are obtained which allow us to produce nontrivial examples of uniform and weak P-ergodic NDMC. Moreover, some category results are also obtained. We stress that all obtained results have potential applications in the classical and non-commutative probabilities.