On Topologically Recurrent Markov Chains

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9409788 Wang In the applications of Markov chains, certain limiting properties are often of interest, such as the convergence rate and the behavior of the limiting distributions. The most general condition of Markov chains related to the above properties is associated with Harris recurrence, which requires the Markov chain returning to every set of positive measure infinitely often, with probability 1. The properties of Harris recurrent Markov chains are well studied. However, there are examples of common Markov chains (e.g., some simple auto-regressive models) which are not Harris recurrent. It appears that Harris recurrence is a strong condition. A.V. Skorokhod suggests another type of recurrence --- topological recurrence. Topological recurrence requires the Markov chain returning to every open set infinitely often with probability 1. Topological recurrence is essentially weaker and more natural than Harris recurrence. Skorokhod's work should have impact in a lot of applications. But so far, this result is still a heoretical advance only. Further understanding of topological recurrence should give us insight into the essence of convergence of Markov chains. The proposed work here is to make a preliminary study of further applications of topological recurrence. ***