Cutoff是指一族遍历 Markov 链收敛到平稳分布时发生急剧变化的现象. 从已有的结果可以知道, cutoff现象的出现与最快强平稳时有关.在本文中, 我们将连续时间单生过程的最快强平稳时用击中时表示,从而得到了单生过程发生cutoff 现象的等价条件.最后, 利用可数状态空间下连续时间 Markov 链的 Kac 公式, 导出击中时和回返时的分布的 Laplace变换之间的关系, 从而实现了回返时的高阶矩可由低阶的击中时的各阶矩线性表出.本文共分为三章.在第1章中, 介绍了关于 cutoff 现象的研究背景和已有成果, 概括了本文的文章脉络和主要结论.在第 2章, 对从 0 出发的可数状态空间连续时间遍历 Markov链,将最快强平稳时用击中时表示出来, 再利用击中时矩的显式表达和已有的cutoff现象发生和最优强平稳时的关系, 得到cutoff现象发生的一个简单的等价条件.在第3章, 我们利用Laplace变化和连续时间Markov 链的Kac 公式,得到了击中时和回返时的分布的 Laplace变换之间的关系,从而实现击中时和回返时各阶矩的相互转化, 用低阶的击中时的矩表示高阶的回返时的矩.

Cutoff phenomenon refers to the gens Markovchain exhibit sharp transition when converge to a stationary distribution. It is known from the existing results, that the cutoffphenomenon associated with the Fastest Strong Stationary Times. in this article, we'll process the Fastest Strong Stationary Times withthe Hitting Times, get a equivalent condition for separation cutoff .This article is divided into three chapters.In chapter 1, we introduce the background of separation cutoff for continuous time single birth processes, and illustrate the mainresults of this article.In chapter 2, we'll process the Fastest Strong Stationary Times with the Hitting Times, get a equivalent condition for separation cutoff .In chapter 3, we use Laplace change and the Kac formula of continuous time Markov chain , We got the relation between the distribution of the hitting time and the return time.