Markov链:遍历性、拟平稳性与不可逆性  

Markov chains: Ergodicity, quasi-stationarity and asymmetry

作　　者：毛永华[1] Yonghua Mao

机构地区：[1]北京师范大学数学科学学院

出　　处：《中国科学：数学》2020年第1期101-118,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11571043和11771047)资助项目

摘　　要：本文以首中时(或回返时)为脉络,从三个方面-Markov链的遍历性、拟平稳分布和不可逆问题-介绍Markov链研究的一些最新进展.这些内容包括:(1)以首中时的矩给出泛函不等式;(2)引入修正的回返时判定各种非常返性;(3)用回返时处理离散时间Markov链的泛函不等式;(4) Markov链首中时的分布表示;(5)以击中时的矩判定一族遍历的Markov过程收敛到平稳分布所产生切断(cutoff)现象;(6)从Markov链生命时的分布找到拟平稳分布存在唯一性;(7)发展Dirichlet原理来判定不可逆Markov链收敛到平稳分布"优于"相应的可逆过程的问题.Based on the first hitting time or return time, we review the development of Markov chain in the study of stationarity, quasi-stationarity and asymmetry. These topics include:(1) using the moments of the return time to derive the functional inequalities;(2) introducing the modified return time to describe the various transience;(3) obtaining the functional inequalities via the return times;(4) using the eigenvalues to describe the distribution of the hitting times;(5) giving the criteria for the cutoff by the hitting times;(6) obtaining the quasi-stationary distribution through the distribution of the life times;(7) developing the Dirichlet principle to judge which is better between the non-reversible Markov chain and its reversible one.

关 键 词：MARKOV链 击中时 回返时 平稳性 拟平稳性 切断 非对称性 

分 类 号：O211.62[理学—概率论与数理统计]