非齐次马氏链的若干遍历性问题
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摘要
概率论是研究大量随机现象统计规律的学科,是许多应用学科的理论基础。诸如信息论、数学风险论、保险精算理论等均是建立在概率论基础之上的。马尔科夫过程更是一类重要的随机过程,它有极为深厚的理论基础,又有广泛的应用空间。马尔科夫链目前已成为内容十分丰富的数学分支。关于齐次马氏链的研究已有相当成熟的结果,并形成了完整的理论体系。与齐次马氏链已取得的丰硕成果相比,非齐次马氏链至今仍是有待深入研究的重要论题。近年来,许多学者对非齐次马氏链的遍历性问题也做了大量研究,如我国学者陈木法院士及杨卫国教授等都对马氏链的各种遍历性进行了一系列的研究。本硕士论文将继续推进这方面的研究。
本文主要研究非齐次马氏链的若干遍历性问题。共分为五章,前两章主要介绍了马氏链的研究背景及与本文相关的马氏链的基本知识和基本引理、定理,为后续章节做好准备。第三章主要是根据不可约、C-强遍历、强遍历、绝对平均强遍历的定义及相关性质,研究了C-强遍历与不可约的关系以及C-强遍历、强遍历、绝对平均强遍历之间的关系,并给出了具体例子。第四章利用矩阵范数的性质、齐次马氏链几何遍历性及非齐次马氏链的相关性质,得到非齐次马氏链绝对平均强遍历的收敛速度。
Probability theory is a subject which concerns the study statistical laws for different kinds of random phenomenon,and theoretica foundations of many applying subjects.Such as Informatin theory,Mathematics Risk theory and Insurance theory for Actuaries etc. Markov process is an important stochastic process. It has profound theoretical fundament,and extensive applied area. At present Markov chains has become very rich content of branch of mathematics. For the study for homogeneous Markov chains, many results have been obtained, which are mature enough to form a complete theoretical system.Compared with the great achievements homogeneous Markov chains has already made, nonhomogeneous Markov chains is still an important topic for futher research now. For ergodicity problem for nonhomogeneous Markov chains, researchers have done much work in recent years, such as chinese researchers Chen and Yang and so on. In this master's thesis,we shall develop this research.
The main purpose of this thesis is to study some problems of ergodic for nonhomogeneous Markov chains.It has five chapters.In chapter one and two,we give an introduction of the basic notions,main results and approaches used in this paper.In chapter three,we mainly research the relationship among irreducible, C-strong ergodicity, absolute mean strong ergodic and strong ergodic through the concrete example.In chapter four, The rate of convergence of absolute mean strong ergodic for nonhomogeneous Markov chains by using the character of norm and nonhomogeneous Markov chains and the geometric ergodic for stationary Markov chains is obtained.
本文主要研究非齐次马氏链的若干遍历性问题。共分为五章,前两章主要介绍了马氏链的研究背景及与本文相关的马氏链的基本知识和基本引理、定理,为后续章节做好准备。第三章主要是根据不可约、C-强遍历、强遍历、绝对平均强遍历的定义及相关性质,研究了C-强遍历与不可约的关系以及C-强遍历、强遍历、绝对平均强遍历之间的关系,并给出了具体例子。第四章利用矩阵范数的性质、齐次马氏链几何遍历性及非齐次马氏链的相关性质,得到非齐次马氏链绝对平均强遍历的收敛速度。
Probability theory is a subject which concerns the study statistical laws for different kinds of random phenomenon,and theoretica foundations of many applying subjects.Such as Informatin theory,Mathematics Risk theory and Insurance theory for Actuaries etc. Markov process is an important stochastic process. It has profound theoretical fundament,and extensive applied area. At present Markov chains has become very rich content of branch of mathematics. For the study for homogeneous Markov chains, many results have been obtained, which are mature enough to form a complete theoretical system.Compared with the great achievements homogeneous Markov chains has already made, nonhomogeneous Markov chains is still an important topic for futher research now. For ergodicity problem for nonhomogeneous Markov chains, researchers have done much work in recent years, such as chinese researchers Chen and Yang and so on. In this master's thesis,we shall develop this research.
The main purpose of this thesis is to study some problems of ergodic for nonhomogeneous Markov chains.It has five chapters.In chapter one and two,we give an introduction of the basic notions,main results and approaches used in this paper.In chapter three,we mainly research the relationship among irreducible, C-strong ergodicity, absolute mean strong ergodic and strong ergodic through the concrete example.In chapter four, The rate of convergence of absolute mean strong ergodic for nonhomogeneous Markov chains by using the character of norm and nonhomogeneous Markov chains and the geometric ergodic for stationary Markov chains is obtained.
引文
1注：渐近循环马氏链的渐近均分割性一文来自Probability in the Engineering and Informational Sciences,24,2010,279-288.
[1]Isaacson, D. and Madsen, R. Markov Chains Theory and Applications[M]. John Wiley & Sons, New York,1976.
[2]杨卫国,李召群.可列非齐次马氏链的绝对平均强遍历性[J].河北煤炭建筑工程学院学报,1996(3).
[3]杨卫国,李芳,王小胜.一类非齐次马氏链的收敛速度[J].江苏大学学报(自然科学版),2005(3).
[4]杨卫国.关于非齐次马氏链的绝对平均强遍历性及若干应用[J].数理统计与应用概率,1996(3).
[5]施仁杰.马尔科夫链基础及应用[M].西安电子科技大学出版社,1992.
[6]Ross, S. Stochastic Processes[M]. John Wily & Sons, New York,1982.
[7]Liu, W. and Yang, W. G.. An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains[J]. Stochastic Process. Appl.,61,129-145, 1996.
[8]Liu, W. and Yang, W. G..The Markov approximation of the sequence of N-valued random variables and a class of small deviation theorems[J]. Stochastic Process. Appl.,89,117-130, 2000.
[9]孟庆生.信息论[M].西安交通大学出版社,1986.
[10]Jarner, S. F. and Roberts, G O., Polynomial convergence rates of Markov chains[J]. Ann. Appl. Probab,12,224-247,2002.
[11]Bowerman, B., David, H. T. and Isaacson, D., The convergence of Cesaro averages for certain nonstationary Markov chains[J]. Stochastic Processes Appl,1977.
[12]Chen, M. F. and Wang, Y. Z., Algebraic convergence of Markov chains[J]. Ann. Appl. Probab,13,604-627,2003.
[13]Huang, C., Isaacson, D. and Vinogrode, B., The rate of convergence of certain nonhomogeneous Markov chains[J]. Z. Wahts cheinlichkeits theorie und verw. Gebeite 35, 1976.
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[16]Yang, Weiguo. Convergence in the Cesaro sense and strong law of large numbers for nonhomogeneous Markov chains[J]. Linear Algebra and its Appl, (345):275-288,2002.
[17]杨卫国,韩金舫.关于非齐次马氏链的Cesaro平均收敛性[J].工程数学学报,14(1)：57-62,1997.
[18]杨卫国,秦忠,庄斌.关于二重非齐次可列马氏链的一类极限定理[J].江苏大学学报(自然科学版),25(1)：52-55,2004.
[19]杨卫国,赵兴平.关于可列非齐次马氏链的强极限定理[J].江苏大学学报(自然科学版),23(1)：49-53,2002.
[20]欧阳光中,姚允龙.数学分析(上)[M].上海：复旦大学出版社,1991.
[21]Zhong, P. P., Yang, W. G and Liang, P. P. The asymptotic equipartition property for asymptoticcircular Markov chains[J]. Probability in the Engineering and Informational Sciences 24:279-288,2010.
[22]严加安.测度论讲义[M].第二版.科学出版社,2004.
[23]汪嘉冈.现代概率论基础.上海：复旦大学出版社,1988.
[24]Paz. A. Introduction to Probabilistie Automate[M]. Aeademie Press, New York,1971.
[25]龚光鲁,钱敏平著.随机过程论.北京：北京大学出版社,2004.
[26]Yang, W. G.. The asymptotic equipartition property for a nonhomogeneous markov Information source[J]. Probab. Eng. Inform. Sci.21,61-66,1998.
[27]McMillan, B.. The basic theorems of information theory[J]. Ann. Math. Statist.24,196-219, 1953.
[28]Barron, A. K. The strong ergodic theorem for densities:Generalized Shannon-McMillan-Breiman theorem[J]. Ann. Probab.13,1292-1303,1985.
[29]Shannon, C.. A mathematical theory of communication. Bell Syst[J]. Tech. J.,27, 379-423,623-656,1948.
[30]Yang, W. G. and Liu, W.. The Asymptotic Equipartition Property for Mth-Order Nonhomogeneous Markov Information Sources[J]. IEEE Transactions on information Theory.50:3326-3330,2004.
[31]Breiman, L.. The individual ergodic theorem of information theory[J]. Ann. Math. Statist. 28,629-635,1957.
[32]Chung, K. L.. The ergodic theorem of information theory[J]. Ann. Math. Statist.32,612-614,1961.
[33]Ye, Z., Berger, T., Ergodic, regularity and asymptotic equipartition property of random fields on trees[J]. Combin. Inform. System Sci.21,157-184,1996.
[34]Liu, Wen, Yang, Weiguo. An extension of Shannon-McMillan Theorem and some limit properties for nonhomogeneous Markov chains[J]. Stochastic Process. Appl. (61):129-145, 1996.
[35]Richard, S. Ellis, Entropy, Large Deviation and Statistical Mechanics[M]. Springer Verlag, New York,1985.
[36]Nicolaie, A.. A generalization of a result concerning the asymptotic behavior of finite Markov chains[J]. Statist. Probab. Lett.78,3321-3329,2008.
[37]Chow, Y. S., Teicher, H.. Probability Theory[M].2nd ed. New York, Springer-Verlag, 1988.
[38]叶中行.信息论基础[M].第二版.高等教育出版社,2007.
[1]Isaacson, D. and Madsen, R. Markov Chains Theory and Applications[M]. John Wiley & Sons, New York,1976.
[2]杨卫国,李召群.可列非齐次马氏链的绝对平均强遍历性[J].河北煤炭建筑工程学院学报,1996(3).
[3]杨卫国,李芳,王小胜.一类非齐次马氏链的收敛速度[J].江苏大学学报(自然科学版),2005(3).
[4]杨卫国.关于非齐次马氏链的绝对平均强遍历性及若干应用[J].数理统计与应用概率,1996(3).
[5]施仁杰.马尔科夫链基础及应用[M].西安电子科技大学出版社,1992.
[6]Ross, S. Stochastic Processes[M]. John Wily & Sons, New York,1982.
[7]Liu, W. and Yang, W. G.. An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains[J]. Stochastic Process. Appl.,61,129-145, 1996.
[8]Liu, W. and Yang, W. G..The Markov approximation of the sequence of N-valued random variables and a class of small deviation theorems[J]. Stochastic Process. Appl.,89,117-130, 2000.
[9]孟庆生.信息论[M].西安交通大学出版社,1986.
[10]Jarner, S. F. and Roberts, G O., Polynomial convergence rates of Markov chains[J]. Ann. Appl. Probab,12,224-247,2002.
[11]Bowerman, B., David, H. T. and Isaacson, D., The convergence of Cesaro averages for certain nonstationary Markov chains[J]. Stochastic Processes Appl,1977.
[12]Chen, M. F. and Wang, Y. Z., Algebraic convergence of Markov chains[J]. Ann. Appl. Probab,13,604-627,2003.
[13]Huang, C., Isaacson, D. and Vinogrode, B., The rate of convergence of certain nonhomogeneous Markov chains[J]. Z. Wahts cheinlichkeits theorie und verw. Gebeite 35, 1976.
[14]Yang, W. G..Convergence in the Cesaro sense and strong law of large numbers for nonhomogeneous Markov chains[J]. Linear Algebra and Its Applications,354,275-28, 2002.
[15]Yang, W. G.. The asymptotic equipartition property for a nonhomogeneous markov Information source[J]. Probab. Eng. Inform. Sci,12,509-518,1998.
[16]Yang, Weiguo. Convergence in the Cesaro sense and strong law of large numbers for nonhomogeneous Markov chains[J]. Linear Algebra and its Appl, (345):275-288,2002.
[17]杨卫国,韩金舫.关于非齐次马氏链的Cesaro平均收敛性[J].工程数学学报,14(1)：57-62,1997.
[18]杨卫国,秦忠,庄斌.关于二重非齐次可列马氏链的一类极限定理[J].江苏大学学报(自然科学版),25(1)：52-55,2004.
[19]杨卫国,赵兴平.关于可列非齐次马氏链的强极限定理[J].江苏大学学报(自然科学版),23(1)：49-53,2002.
[20]欧阳光中,姚允龙.数学分析(上)[M].上海：复旦大学出版社,1991.
[21]Zhong, P. P., Yang, W. G and Liang, P. P. The asymptotic equipartition property for asymptoticcircular Markov chains[J]. Probability in the Engineering and Informational Sciences 24:279-288,2010.
[22]严加安.测度论讲义[M].第二版.科学出版社,2004.
[23]汪嘉冈.现代概率论基础.上海：复旦大学出版社,1988.
[24]Paz. A. Introduction to Probabilistie Automate[M]. Aeademie Press, New York,1971.
[25]龚光鲁,钱敏平著.随机过程论.北京：北京大学出版社,2004.
[26]Yang, W. G.. The asymptotic equipartition property for a nonhomogeneous markov Information source[J]. Probab. Eng. Inform. Sci.21,61-66,1998.
[27]McMillan, B.. The basic theorems of information theory[J]. Ann. Math. Statist.24,196-219, 1953.
[28]Barron, A. K. The strong ergodic theorem for densities:Generalized Shannon-McMillan-Breiman theorem[J]. Ann. Probab.13,1292-1303,1985.
[29]Shannon, C.. A mathematical theory of communication. Bell Syst[J]. Tech. J.,27, 379-423,623-656,1948.
[30]Yang, W. G. and Liu, W.. The Asymptotic Equipartition Property for Mth-Order Nonhomogeneous Markov Information Sources[J]. IEEE Transactions on information Theory.50:3326-3330,2004.
[31]Breiman, L.. The individual ergodic theorem of information theory[J]. Ann. Math. Statist. 28,629-635,1957.
[32]Chung, K. L.. The ergodic theorem of information theory[J]. Ann. Math. Statist.32,612-614,1961.
[33]Ye, Z., Berger, T., Ergodic, regularity and asymptotic equipartition property of random fields on trees[J]. Combin. Inform. System Sci.21,157-184,1996.
[34]Liu, Wen, Yang, Weiguo. An extension of Shannon-McMillan Theorem and some limit properties for nonhomogeneous Markov chains[J]. Stochastic Process. Appl. (61):129-145, 1996.
[35]Richard, S. Ellis, Entropy, Large Deviation and Statistical Mechanics[M]. Springer Verlag, New York,1985.
[36]Nicolaie, A.. A generalization of a result concerning the asymptotic behavior of finite Markov chains[J]. Statist. Probab. Lett.78,3321-3329,2008.
[37]Chow, Y. S., Teicher, H.. Probability Theory[M].2nd ed. New York, Springer-Verlag, 1988.
[38]叶中行.信息论基础[M].第二版.高等教育出版社,2007.