可列m重非齐次马氏链的强大数定律
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摘要
马氏链作为描述一类实际问题的数学模型，在经济学、生命科学、随机服务系统、计算科学、随机分形等邻域中取得了极为丰硕的成果。近几十年来，人们对非齐次马氏链的极限定理和遍历性开展了大量研究。多重马氏链的概念是一般马氏链概念的自然推广，多重马尔可夫信源是一类很重要的信源，如语声、电视信号。故对多重马氏链理论方面的研究具有很大的研究意义。
本文的目的是研究可列m重非齐次马氏链的遍历性及强大数定律。第一章主要介绍马氏链的相关研究及进展。第二章介绍马氏链的基础理论知识。第三章在第二章的基础上给出可列m重非齐次马氏链的定义及相关定义与性质。第四章研究可列m重非齐次马氏链的强大数定律。首先利用鞅论中的结果与遍历系数相结合的方法得到了可列m重非齐次马氏链关于m+1元函数的一个强大数定律；其次在已有结论的基础上，对数列{a_n，n≥m}加以限制，得到可列m重非齐次马氏链泛函的一个强大数定律，并推广刘国欣等人关于可列非齐次马氏链强大数定律中的一些结果。第五章首先引用杨卫国对马氏链绝对平均强遍历的概念，利用转移概率的形式给出可列m重非齐次马氏链绝对平均强遍历的概念，并给出其充分条件，最后讨论其在马氏决策过程和信息论中的应用。
Markov chain is a mathematic model describing practical problems,it has got rich progress in many areas such as economics,biology,stochastic service system,computer science. The research about limit theorems and ergodic properties has been researching in recent years. The definition of mth-order nonhomogeneous Markov chain is an extension of the definition of Markov chain. Mth-order Markov information source is an important information source .So the research about the theory of mth-order Markov chain is very important.
The article is going to study the ergodic properties and strong law of large numbers about mth-order countable nonhomogeous 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 subsequent chapters. In the third chapter,we introduce the definition of mth-order countable nonhomogeneous Markov chains on the basis of the second chapter. In the fourth chapter,we study the strong law of large numbers for mth-order countable nonhomogeneous Markov chains .Firstly,we get a strong law of large numbers of the functions of m+1th variables of mth-order countable nonhomogeneous Markov chains by means of martingable method and the ergodic coefficient. Then constraining number sequence {a_n,n≥m}, we obtain a strong law of large numbers for functionals of countable nonhomogeneous Markov chains based on obtained results. It is an extension of Liu Guoxin's result. In the fifth chapter,firstly we quote the concept of absolute average strong ergodic which Yang Weiguo has introduced,then we introduce the concept of absolute average strong ergodic in the form of transition probability and give its sufficient condition.Finally,we discuss its application in Markov function system and information theory.
本文的目的是研究可列m重非齐次马氏链的遍历性及强大数定律。第一章主要介绍马氏链的相关研究及进展。第二章介绍马氏链的基础理论知识。第三章在第二章的基础上给出可列m重非齐次马氏链的定义及相关定义与性质。第四章研究可列m重非齐次马氏链的强大数定律。首先利用鞅论中的结果与遍历系数相结合的方法得到了可列m重非齐次马氏链关于m+1元函数的一个强大数定律；其次在已有结论的基础上，对数列{a_n，n≥m}加以限制，得到可列m重非齐次马氏链泛函的一个强大数定律，并推广刘国欣等人关于可列非齐次马氏链强大数定律中的一些结果。第五章首先引用杨卫国对马氏链绝对平均强遍历的概念，利用转移概率的形式给出可列m重非齐次马氏链绝对平均强遍历的概念，并给出其充分条件，最后讨论其在马氏决策过程和信息论中的应用。
Markov chain is a mathematic model describing practical problems,it has got rich progress in many areas such as economics,biology,stochastic service system,computer science. The research about limit theorems and ergodic properties has been researching in recent years. The definition of mth-order nonhomogeneous Markov chain is an extension of the definition of Markov chain. Mth-order Markov information source is an important information source .So the research about the theory of mth-order Markov chain is very important.
The article is going to study the ergodic properties and strong law of large numbers about mth-order countable nonhomogeous 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 subsequent chapters. In the third chapter,we introduce the definition of mth-order countable nonhomogeneous Markov chains on the basis of the second chapter. In the fourth chapter,we study the strong law of large numbers for mth-order countable nonhomogeneous Markov chains .Firstly,we get a strong law of large numbers of the functions of m+1th variables of mth-order countable nonhomogeneous Markov chains by means of martingable method and the ergodic coefficient. Then constraining number sequence {a_n,n≥m}, we obtain a strong law of large numbers for functionals of countable nonhomogeneous Markov chains based on obtained results. It is an extension of Liu Guoxin's result. In the fifth chapter,firstly we quote the concept of absolute average strong ergodic which Yang Weiguo has introduced,then we introduce the concept of absolute average strong ergodic in the form of transition probability and give its sufficient condition.Finally,we discuss its application in Markov function system and information theory.
引文
[1] David Freedman Markov Chains, Springer-Verlag, New York, 1983
[2] John G. Kemeny, J. Laurie Snell, Finite Markov Chains, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1976
[3] 施仁杰 马尔科夫链基础及应用，西安电子科技大学出版社，1992
[4] 钱敏平，龚光鲁著 应用随机过程，北京大学出版社，1998
[5] 胡迪鹤 随机过程论——基础．理论．应用，武汉大学出版社，2000
[6] 朱成熹等 非齐次马尔科夫链函数的强大数定律，数学学报，1988，31(4)：465-474
[7] Rosenbllat R M. Some theorems concerning the strong law of numbers for nonhomogeneous Markov chains. Ann Math Statist, 1964, 35; 566-576
[8] Liu Guoxin, Liu Wen. On the strong law of large numbers for functionals of. countablenonhomogeneous Markov chains. Stochastic Processes Appl, 1994, 50:375-391.
[9] 刘文，杨卫国．一类对可列非齐次Markov链普遍成立的强大数定律．科学通报，1992，37(16)：1448-1451
[10] 刘文，杨卫国．可列非齐次马氏链的若干极限定理．应用数学学报，1992，15(4)：479-489
[11] D. Isaacson, R. Madsen, Markov chains theorys and application, 1976, Wiley, New York
[12] C. Huang, D. Isaacson and B. Binograde, The rate of convergence of certain nonhomogeneous Markov chains, Z. Warhrcheinlichbeitstheorie and Verw. Geberte 35(1976) 141-146
[13] B. E. RHOADES, The convergence of matrix transforms for certain Markov chains, Stochastic Processes and their Applications 9(1979)85-93
[14] 陈永义，傅自晦，张学显 非齐次马尔科夫链遍历性的一些结果，系统科学与数学，1996年10月，Vol．16，No．4，311-317
[15] B. Bowerman, H. T. David and D. Isaacson, The convergence of Cesaro averages for certain nonstationary Markov chains, Stochastic Processes and their Applications 5(1977)221-230
[16] 杨卫国 关于非齐次马氏链的绝对平均强遍历性及应用，数理统计与应1996，Vol．11，No．3(220-226)
[17] 杨卫国，陈璞玉 关于非齐次马氏链的绝对平均强遍历性，河北煤炭建筑工程学报，1994，No．4(30-36)
[18] Weiguo Yang and Wen Liu, The Asympttic Equipartition Property for Mth-Order Nonhomogeneous Markov Information Sources. IEEE Transactions on information Theory2004, 50(12): 1-6
[19] 汪忠志，杨卫国．二重非齐次马氏链及其随机变换的若干强极限定理．系统科学与学，2004，24(4)：451-462
[20] Yang W G. Convergence in the cesaro sense and strong law of large numbers for nonhomogeneous Markov chains. Linear Algebra and its Appl, 2002(354): 275-288
[21] Yan J A, Liu W, Yang W G. A limit theorem for partial sums of random variables and its application. Stats Probab. Letts. 2003, 62: 79-86
[22] 王小胜，杨卫国，郭海英．关于随机变量序列的强大数定律．数学的实践与认识．2005，35(1)：215-218
[23] Wen Liu, Weiguo Yang. A class of strong limit theorems for the sequences of arbitrary random variables. Statistices and Probability Letters, 2003, 64: 121-131
[24] Paz, A. Introduction to Probabilistic Automate, Academic Press,(1971), New York.
[25] Cover, T. M. and Thomas, J. A., Elements of Information Theory, Wiley, New York, 1991
[2] John G. Kemeny, J. Laurie Snell, Finite Markov Chains, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1976
[3] 施仁杰 马尔科夫链基础及应用，西安电子科技大学出版社，1992
[4] 钱敏平，龚光鲁著 应用随机过程，北京大学出版社，1998
[5] 胡迪鹤 随机过程论——基础．理论．应用，武汉大学出版社，2000
[6] 朱成熹等 非齐次马尔科夫链函数的强大数定律，数学学报，1988，31(4)：465-474
[7] Rosenbllat R M. Some theorems concerning the strong law of numbers for nonhomogeneous Markov chains. Ann Math Statist, 1964, 35; 566-576
[8] Liu Guoxin, Liu Wen. On the strong law of large numbers for functionals of. countablenonhomogeneous Markov chains. Stochastic Processes Appl, 1994, 50:375-391.
[9] 刘文，杨卫国．一类对可列非齐次Markov链普遍成立的强大数定律．科学通报，1992，37(16)：1448-1451
[10] 刘文，杨卫国．可列非齐次马氏链的若干极限定理．应用数学学报，1992，15(4)：479-489
[11] D. Isaacson, R. Madsen, Markov chains theorys and application, 1976, Wiley, New York
[12] C. Huang, D. Isaacson and B. Binograde, The rate of convergence of certain nonhomogeneous Markov chains, Z. Warhrcheinlichbeitstheorie and Verw. Geberte 35(1976) 141-146
[13] B. E. RHOADES, The convergence of matrix transforms for certain Markov chains, Stochastic Processes and their Applications 9(1979)85-93
[14] 陈永义，傅自晦，张学显 非齐次马尔科夫链遍历性的一些结果，系统科学与数学，1996年10月，Vol．16，No．4，311-317
[15] B. Bowerman, H. T. David and D. Isaacson, The convergence of Cesaro averages for certain nonstationary Markov chains, Stochastic Processes and their Applications 5(1977)221-230
[16] 杨卫国 关于非齐次马氏链的绝对平均强遍历性及应用，数理统计与应1996，Vol．11，No．3(220-226)
[17] 杨卫国，陈璞玉 关于非齐次马氏链的绝对平均强遍历性，河北煤炭建筑工程学报，1994，No．4(30-36)
[18] Weiguo Yang and Wen Liu, The Asympttic Equipartition Property for Mth-Order Nonhomogeneous Markov Information Sources. IEEE Transactions on information Theory2004, 50(12): 1-6
[19] 汪忠志，杨卫国．二重非齐次马氏链及其随机变换的若干强极限定理．系统科学与学，2004，24(4)：451-462
[20] Yang W G. Convergence in the cesaro sense and strong law of large numbers for nonhomogeneous Markov chains. Linear Algebra and its Appl, 2002(354): 275-288
[21] Yan J A, Liu W, Yang W G. A limit theorem for partial sums of random variables and its application. Stats Probab. Letts. 2003, 62: 79-86
[22] 王小胜，杨卫国，郭海英．关于随机变量序列的强大数定律．数学的实践与认识．2005，35(1)：215-218
[23] Wen Liu, Weiguo Yang. A class of strong limit theorems for the sequences of arbitrary random variables. Statistices and Probability Letters, 2003, 64: 121-131
[24] Paz, A. Introduction to Probabilistic Automate, Academic Press,(1971), New York.
[25] Cover, T. M. and Thomas, J. A., Elements of Information Theory, Wiley, New York, 1991