On Topologically Recurrent Markov Chains

论拓扑循环马尔可夫链

• 批准号: 9409788
• 负责人: Mei Wang
• 金额: $ 1.8万
• 依托单位: Illinois Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-01 至 1995-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9409788&HistoricalAwards=false
• 关键词: Topologically; Recurrent; Markov; Chains

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. ***

9409788王在马尔可夫链的应用中，某些限制属性通常是感兴趣的，例如收敛速率和限制分布的行为。与上述特性相关的马尔可夫连锁店的最一般条件与哈里斯复发有关，哈里斯复发需要返回到每一套正措施，经常经常返回每一套正措施，概率为1。 但是，有一些常见的马尔可夫链（例如，一些简单的自动回归模型）的例子不是哈里斯经常性。 似乎哈里斯复发是强烈的状况。 A.V. Skorokhod提出了另一种类型的复发---拓扑复发。 拓扑复发需要马尔可夫链返回到每个开放式的概率1。拓扑复发本质上比哈里斯复发更弱，更自然。 Skorokhod的工作应该在许多应用程序中产生影响。但是到目前为止，这一结果仍然是一个遥远的进步。 对拓扑复发的进一步理解应该使我们深入了解马尔可夫链融合的本质。 拟议的工作是对拓扑复发的进一步应用进行初步研究。 ***