离散时间Markov链的几何遍历速度  

Convergence rates in geometric ergodicity for discrete-time Markov chains

作　　者：宋延红[1] Yanhong Song

机构地区：[1]中南财经政法大学统计与数学学院,武汉430073

出　　处：《中国科学：数学》2020年第8期1087-1096,共10页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11501576);中南财经政法大学研究生教育创新计划(批准号:201821301)资助项目。

摘　　要：本文研究一般状态空间离散时间几何遍历Markov链的收敛速度.当状态空间中含有原子时,本文采用与可数状态空间类似的方法,即采用谱理论法研究几何遍历速度.对于不含原子的Markov链,先利用Nummelin分裂技术构造出含有原子的新Markov链,再建立新Markov链与原Markov链几何遍历速度的关系,最后由最小化条件和首次回返时的几何阶矩得到原Markov链几何遍历的收敛速度.进一步,本文将所得结果应用于研究Metropolis-Hastings算法.In this paper,we investigate the convergence rates in geometric ergodicity for discrete-time Markov chains on general state spaces.When the state space contains an atom,we study the convergence rates by the spectral theory,which is similar to that on countable state spaces.However,atoms are less frequent.For the general case,we first use the Nummelin splitting technique to construct a new Markov chain which contains an atom,and then establish the connection of the geometrically ergodic rates between the new and the original Markov chains.We also obtain the convergence rate in geometric ergodicity for the original Markov chain by the minorization condition and the geometric moment of the first return time.Moreover,we apply the main result to study the Metropolis-Hastings algorithm.

关 键 词：MARKOV链 几何遍历 收敛速度 

分 类 号：O18[理学—数学]