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Quasi-Feller Markov chains. (English) Zbl 0981.60063
Summary: We consider the class of Markov kernels for which the weak or strong Feller property fails to hold at some discontinuity set. We provide a simple necessary and sufficient condition for existence of an invariant probability measure as well as a Foster-Lyapunov sufficient condition. We also characterize a subclass, the quasi-weak (or quasi-strong) Feller kernels, for which the sequences of expected occupation measures share the same asymptotic properties as for (weak or strong) Feller kernels. In particular, it is shown that the sequences of expected occupation measures of strong and quasi-strong Feller kernels with an invariant probability measure converge setwise to an invariant measure.

60J05 Discrete-time Markov processes on general state spaces
60B10 Convergence of probability measures
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