On Topologically Recurrent Markov Chains

[409788]在马尔可夫链的应用中，一些极限性质经常引起人们的兴趣，如收敛速度和极限分布的性质。与上述性质相关的马尔可夫链的最一般条件与Harris递归有关，Harris递归要求马尔可夫链以1的概率无限次地返回到每一组正测度。研究了Harris递归马尔可夫链的性质。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）不是哈里斯递归的。看来哈里斯复发是一个很强的条件。A.V. Skorokhod提出了另一种递归——拓扑递归。拓扑递归要求马尔可夫链以1的概率无限次地返回到每一个开集。拓扑递归本质上比哈里斯递归更弱、更自然。Skorokhod的工作应该会对很多应用产生影响。但到目前为止，这一结果还只是理论上的进步。对拓扑递归的进一步理解将使我们深入了解马尔可夫链收敛的本质。本文提出的工作是对拓扑递归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. ***