Approximations of non-homogeneous Markov chains on abstract states spaces

Approximations of nonhomogeneous discrete Markov chains (NDMC) play an essential role in both probability and statistics. In all these settings, it is crucial to consider random variables in appropriate spaces. Therefore, the abstract considerations of such spaces lead to investigating the approximations in ordered Banach space scheme. In this paper, we consider two topologies on the set of NDMC of abstract state spaces. We establish that the set of all uniformly P-ergodic NDMC is norm residual in NDMC. The set of point-wise weak P-ergodic NDMC is also considered and such sets are shown to be a Gδ-subset (in strong topology) of NDMC. We point out that all the deduced results are new in the classical and non-commutative probabilities, respectively, since in most of earlier results the limiting projection is taken as a rank one projection. Indeed, the obtained results give new insight into data-analysis and statistics.