Spectral Gap and Convergence Rate for Discrete-time Markov Chains  

作　　者：Yong Hua MAO Yan Hong SONG 

机构地区：[1]School of Mathematical Sciences,LMCS,Ministry of Education,Beijing Normal University

出　　处：《Acta Mathematica Sinica,English Series》2013年第10期1949-1962,共14页数学学报（英文版）

基　　金：Supported in part by 985 Project,973 Project(Grant No.2011CB808000);National Natural Science Foundation of China(Grant No.11131003);Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20100003110005);the Fundamental Research Funds for the Central Universities

摘　　要：Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2（π）-space, denoted by gap（P）, is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap（P） and the convergence rate of P^n. When P is transient, the convergence rate of pn is equal to 1 - gap（P）. When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of pn in terms of gap（P）. These results are extended to L^∞ （π）-space.Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2（π）-space, denoted by gap（P）, is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap（P） and the convergence rate of P^n. When P is transient, the convergence rate of pn is equal to 1 - gap（P）. When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of pn in terms of gap（P）. These results are extended to L^∞ （π）-space.

关 键 词：Spectral gap convergence rate geometric ergodicity TRANSIENCE strong ergodicity uni-form decay 

分 类 号：O211.62[理学—概率论与数理统计]