On Topologically Recurrent Markov Chains

9409788王在马尔可夫链的应用中，某些极限性质经常是人们感兴趣的，例如极限分布的收敛速度和行为。与上述性质相关的马氏链的最一般条件是与Harris常返性有关的，它要求马氏链以1的概率无限地返回到每一正测度集。然而，有一些常见的马尔可夫链的例子(例如，一些简单的自回归模型)不是Harris循环的。看来Harris常返性是一个很强的条件。A.V.斯科罗霍德提出了另一种类型的递归-拓扑递归。拓扑常返性要求马尔可夫链以1的概率无限地返回到每个开集。拓扑常返性本质上比Harris常返性更弱、更自然。斯科罗霍德的工作应该会在很多应用中产生影响。但到目前为止，这一结果仍然只是一个理论上的进步。通过对拓扑常返性的进一步理解，我们可以更深入地了解马尔可夫链的收敛本质。本文提出的工作是对拓扑常返性的进一步应用进行初步研究。***

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. ***