我们用几何方法研究可数状态空间非时齐马氏链遍历理论, 将 Hajnal不等式推广到可数状态空间, 得到不等式更为一般的形式. 借助单纯形理论和 Dobrushin 遍历系数等工具证明了推广的不等式. 本文主要分为四部分. 第一部分是综述, 介绍了本文的研究背景, 有限状态空间 Hajnal不等式的推广形式及理论支撑; 另外还介绍了本文需要的一些预备知识和主要结论. 第二部分给出可数状态状态空间经典的 Hajnal 不等式#以及该不等式分析方法的证明. 第三部分用单纯形理论给出非时齐马氏链遍历的几何证明, 进而得到了可数状态空间 Hajnal不等式最一般的形式, 使得仍有 \ ${\rm diam}(PQ)\leq(1-\lambda(P)){\rm diam}(Q)$ 成立. 最后一部分使用 Dobrushin遍历系数角度证明了\ ${\rm diam}_{1}(PQ)\leq(1-\lambda(P)){\rm diam}_{1}(Q)$.

We use the geometric approach to study the classical problem of weakly ergodic non-homogeneous Markov chains on countable state space. We establish a more general form of classical Hajnal inequality. And the generalized inequality is proved via simplices theory and Dobrushin ergodic coefficient. This thesis consists of four parts. The first part presents a summary of this thesis, which includes the research background, the generalized Hajnal inequality and its theoretical support of non-homogeneous Markov chains on finite states and some preliminary knowledge. The second part introduces the classical Hajnal inequality: \ ${\rm diam}_{\infty}(PQ)\leq(1-\lambda(P)){\rm diam}_{\infty}(Q)$. And we recall an analysis approach. In section 3, using simplices theory, we show weakly ergodicity for non-homogeneous Markov chains and a more general inequality on infinite state space. In the last section, we use Dobrushin ergodic coefficient to proof the inequality:\ ${\rm diam}_{1}(PQ)\leq(1-\lambda(P)){\rm diam}_{1}(Q)$, which shows that the result is correct.