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 Wang 在马尔可夫链的应用中，某些限制性质常常令人感兴趣，例如收敛速度和限制分布的行为。与上述性质相关的马尔可夫链最一般的条件与 Harris 递归相关，这要求马尔可夫链以概率 1 无限频繁地返回到每组正测度。Harris 递归马尔可夫链的性质已经得到了很好的研究。 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强条件。 AV Skorokhod 提出了另一种类型的复发——拓扑复发。 拓扑递推要求马尔可夫链以概率 1 无限频繁地返回到每个开集。拓扑递推本质上比哈里斯递推更弱、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，这一结果仍然只是理论上的进展。 对拓扑递推的进一步理解应该能让我们深入了解马尔可夫链收敛的本质。 这里提出的工作是对拓扑递推的进一步应用进行初步研究。 ***