【作者】 李芳；

【导师】 杨卫国；

【作者基本信息】 江苏大学 ， 应用数学， 2005， 硕士

【摘要】 马尔科夫过程是一类重要的随机过程,它有极为深厚的理论基础,如拓扑学、函数论、泛函分析、近世代数和几何学,又有广泛的应用空间,如物理、化学、生物、天文、计算机、通信、经济管理等等众多领域。有关齐次马氏链的研究,已形成了较完整的理论体系;关于非齐次马氏链的研究,人们一直在陆续进行中。本文主要研究非齐次马氏链的收敛及收敛速度。 本文第一章主要介绍马氏链的相关研究及进展。第二章介绍后续章节所需用到了基础理论知识。第三章研究一类非齐次马氏链的收敛速度及绝对平均强遍历性。首先在B.Bowerman等人研究转移矩阵列收敛的一类非齐次马氏链,其Cesaro平均收敛的收敛速度基础上,研究转移矩阵列平均收敛到一周期强遍历随机矩阵的一类非齐次马氏链,通过控制转移矩阵列平均收敛的收敛速度,利用矩阵范数的性质、非齐次马氏链的相关性质等,得到该非齐次马氏链转移矩阵Cesaro平均收敛的收敛速度,是B.Bowerman等人结果的一个推广,并将这一结果应用到期望平均费用中;其次引用杨卫国提出马氏链绝对平均强遍历的概念,给出齐次马氏链绝对平均强遍历与强遍历的等价性,然后通过引进另一个强遍历的非齐次马氏链,给出一个非齐次马氏链绝对平均强遍历的充分条件,是杨卫国作出一个非齐次马氏链绝对平均强遍历的充分条件的推广。第四章通过引进一正规矩阵,证明了一个关于非齐次马氏链的收敛定理,是杨卫国等人的一个关于非齐次马氏链Cesaro平均收敛定理的推广。第五章作一个简单总结。更多还原

【Abstract】 Markov processes is an important probabilistic processes.It has profound theoretic foundament ,such as topology,theory of functions,functional analysis,modem algebra and geometry,and it has extensive applied area,such as physics, chemistry, biology, astronomy, computer,communication,management of economy.The research about homogeneous Markov chains has formed integrated theoretic system;The research about nonhomogeneous Markov chain has been researching.This article is going to study the convergence and rate of convergence about nonhomogeneous Markov chains.In the first chapter,we introduce the research and progresses about Markov chains.In the second chapter,we introduce the basic theory which needs to use in the subsequent chapters.In the third chapter,we study the rate of convergence of nonhomogeneous Markov chains and the absolute average strong ergodic.Firstly,based on the B.Bowerman’s result about the rate of convergence in Cesaro sense of certain nonhomogeneous Markov chains which the transition matrices converge,we are to study a certain nonhomogenous Markov chains which the transition matrices average converge to a period strongly ergodic stochastic matrice,and control the average convergenc rate of transition matrices,then we get the rate of convergence in Cesaro sense about the nonhomogeneous Markov chains by used the character of norm and the character of nonhomogeneous Markov chains.It is an extension of a B.Bowerman’s result.We also discuss the application on the expected average cost.And then we quote the concept of absolute average strong ergodic which YangWeiguo has introduced.We give the equivalence between the absolute average strong ergodic and strong ergodic for homogeneous Markov chains.And we give sufficient condition of absolute average strong ergodic for a nonhomogeneous Markov chains through injecting another nonhomogeneous Markov chains which is strong ergodic,which is an extension of YangWeiguo’s result aboutabsolute average strong ergodic for a nonhomogeneous Markov chains.In the fourth chapter,we quoted a normal matrix,and proved a convergence theorem about nonhomogeneous Markov chains.The theorem is an extension of Yang Weiguo’s Cesaro average convergence theorem of nonhomengeous Markov chains.更多还原