一类有限非齐次马氏链的强极限定理及应用
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摘要
马尔可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。近几十年来,人们对马氏链的极限定理、遍历性和熵率等信息度量的相关性质开展了大量研究。本文主要研究一类非齐次马氏链的强极限定理,计算中国象棋的熵率和预测汇率。
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的基础理论知识。第三章主要给出非齐次马氏链的一类极限定理。在杨卫国,刘文对非齐次马氏信源的渐近均分割性研究的基础上,给出非齐次马氏链的一类极限定理,作为主要结果的推论,得到非齐次马氏信源的相对熵密度的极限性质。第四章主要通过对加权图上的随机游动的熵率的研究,引进了中国象棋各棋子的熵率,从而可以比较中国象棋各棋子的自由度。第五章分析了汇率风险的类型以及衡量要素,运用马尔科夫链的理论预测汇率,利用相对强弱指数这个指标进行决策,并通过实例检验,证明了这个模型的可行性和实用性。
Markov process is an important probabilistic process.It has profound theoretic foundation,such as topology,theory of functions, functional analysis,modern algebra and geometry.In addition,it has extensive applied area,such as physics,chemistry,biology,astronomy, computer,communication and management of economy.The research about homogeneous Markov chains has formed integrated theoretic system.The research about limit theorems and ergodic properties has been researching in recent years.This article is going to study limit theorems for finite non-homogeneous Markov chains,calculating entropy rate and forecasting for exchange rate.
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 limit theorems for finite non-homogeneous Markov chains.Firstly, Yang weiguo and Liu Wen studied the asymptotic equipartition properties for Markov information,based on the uniform integrability of random variable sequences,we are to give limit theorems..As corollary,we obtain the limit properties for entropy density for non-homogeneous Markov.In the fourth chapter,we study the entropy rate of a random walk on a weighted graph,introduce the entropy rate of a chess about china chess,then compare to the freedom of a chess about china chess.In the last chapter,we analyse the type of the exchange rate risk and the judged elements.We forecast for exchange rate with the theory of Markov chains and make a strategic decision with reference to the powerful-weak index.At last,we prove the feasibility and the practicability by force of testing the example.
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的基础理论知识。第三章主要给出非齐次马氏链的一类极限定理。在杨卫国,刘文对非齐次马氏信源的渐近均分割性研究的基础上,给出非齐次马氏链的一类极限定理,作为主要结果的推论,得到非齐次马氏信源的相对熵密度的极限性质。第四章主要通过对加权图上的随机游动的熵率的研究,引进了中国象棋各棋子的熵率,从而可以比较中国象棋各棋子的自由度。第五章分析了汇率风险的类型以及衡量要素,运用马尔科夫链的理论预测汇率,利用相对强弱指数这个指标进行决策,并通过实例检验,证明了这个模型的可行性和实用性。
Markov process is an important probabilistic process.It has profound theoretic foundation,such as topology,theory of functions, functional analysis,modern algebra and geometry.In addition,it has extensive applied area,such as physics,chemistry,biology,astronomy, computer,communication and management of economy.The research about homogeneous Markov chains has formed integrated theoretic system.The research about limit theorems and ergodic properties has been researching in recent years.This article is going to study limit theorems for finite non-homogeneous Markov chains,calculating entropy rate and forecasting for exchange rate.
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 limit theorems for finite non-homogeneous Markov chains.Firstly, Yang weiguo and Liu Wen studied the asymptotic equipartition properties for Markov information,based on the uniform integrability of random variable sequences,we are to give limit theorems..As corollary,we obtain the limit properties for entropy density for non-homogeneous Markov.In the fourth chapter,we study the entropy rate of a random walk on a weighted graph,introduce the entropy rate of a chess about china chess,then compare to the freedom of a chess about china chess.In the last chapter,we analyse the type of the exchange rate risk and the judged elements.We forecast for exchange rate with the theory of Markov chains and make a strategic decision with reference to the powerful-weak index.At last,we prove the feasibility and the practicability by force of testing the example.
引文
[1]S.P.Meyn,R.L.Tweedie Markov chains and Stochastic Stability(1993) Springer-Verlag
[2]David Freedman,Markov Chains,Springer-Verlag[M],New York,1983
[3]John G.Kemeny,J.Laurie Snell,Finite,Markov Chains,Springer-Verlag,New York Berlin Heidelberg Tokyo,1976
[4]D.Isaacson,R.Madsen,Markov chains theorys and application[M],1976,Wiley,New York
[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][美]钟开莱 概率论教程[M],刘文等译,上海科学技术出版社,1989
[7]朱成熹,随机极限引论[M],天津:南开大学出版社,1987
[8]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
[9]Rosenbllat R M.Some theorems concerning the strong law of numbers for non-homogeneous Markov chains.Ann Math Statist,1964,35;566-576
[10]Liu Guoxin,Liu Wen.On the strong law of large numbers for functionals of countable non-homogeneous Markov chains.Stochastic Processes Appl,1994,50:375-391
[11]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
[12]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
[13]杨卫国,刘开弟,董卫 关于可列非齐次马氏链Cesaro平均收敛性及二元函数的强大数定律,数学物理学报,1998,18(增刊),27-34
[14]C.E.Shannon.A mathematical theory of communication.Bell Syst.Tech.J.,27:379-423,623-656,1948.
[15]B.McMillan.The basic theorems of information theory.Ann.Math.Stat.,24:196-219,1953.[16]L.Breiman.The individual ergodic theorems of information theory.Ann.Math.Stat.,28:809-811,1957.With correction made in 31:809-810.
[17]S K.L.Chung.A note on the ergodic theorem of information theory.Ann.Math.Stat.,32:612-614,1961.
[18].C.Moy.Generalization of the Shonnon-McMillan theorem.Pacific J.Math,pages 705-714,1961.
[19]J.R.Pierce.The early days of information theory.IEEE Trans.Inf.Theory,pages 3-8,Jan.1973.
[20]J.C.Kieffer.A simple proof of the Moy-Perez.generalization of the Shannon-McMillan theorem.Pacific J.Math.,51:203-206,1974.
[21]A.R.Barron.The strong ergodic theorem for densities:generalized Shannon- McMillan-
Breiman theorem.Ann.Prob.,13:1292-1303,1985.
[22]S.Orey.On the Shannon-Perez-Moy theorem.Contemp.Math.,41:319-327,1985.
[23]P.Algoet and T.M.Cover.A sandwich proof of the Shannon- McMillan- Breiman theorem.Ann.Prob.,16(2):899-909,1988.
[24]杨卫国,非齐次马氏链熵率存在定理[J].数学的实践与认识,1993,No.2:86-90.
[25]刘文,杨卫国.相对熵密度与任意二进信源的若干极限性质.应用数学学报.Vol.17 No.11994.
[26]刘文,杨卫国.关于有限马氏链相对熵密度和随机条件熵的一类极限定理,应用数学学报,1995,Vol.18,No.2(234-247)
[27]W.,Liu,W.C.,Yang,A limit theorem for the entropy density of nonhomogeneous Markov information source[J].Statistics & Probability Letters,1995,22:295-301
[28]W.G.,Yang,W.,Liu,The asymptotic equipartition property for Mth-order nonhomogeneous Markov information sources[J].IEEE TRANSANCTIONS ON INFORMATION TEORY,2004,Vol.50,NO.12:3326-3330
[29]刘文,杨卫国.关于非齐次马氏信源的渐进均匀分割性.应用概率统计.Vol.13 No.41997
[30]Yang Weiguo,Liu Wen.Strong law of large numbers and Shannon-McMillan theorem for Markov chains field on Cayley tree[J],Acta Mathmatica Scientia,2001,21B(4):495-502
[31]刘文,王丽英,杨卫国.二进树上奇偶马氏链场的若干强极限定理与Shannon-McMillan定理的一种逼近,应用概率统计,Vol.18,277-285,2002
[32]杨卫国,吴小太,王豹.一类隐马尔可夫模型的若干极限性质[J].江苏大学学报:自然科学版,2006,27(5):467-468.
[33]吴小求.证券投资学[M].北京:中国人民大学出版社.2004.310-321
[34]汪嘉冈。现代概率论基础[M]。上海:复旦大学出版社,1988。
[35]L.,Cover,J.,Thomas,Elements of Information Theory[M].Wiley,New York,1991
[36]Donald R.Van Deventer,Kenji lmia,Mark Mesler.高级金融风险管理[M]译者朱世武,邢丽.中国人民大学出版社,2006
[37]段文斌,桂佳.外汇市场与外汇业务[M].经济管理出版社,2001年
[38]彭志行,夏乐天.马尔科夫链及其在股市分析中的应,应用数学,2004(17)增159—163期
[39]张静,江寿阳.人民币均衡汇率与中国外贸[M].高等教育出版社,2005
[40]Yang Weiguo Strong limit theorems for arbitrary.J.Math.Anal.Appl.326(2007)1445-1451
[2]David Freedman,Markov Chains,Springer-Verlag[M],New York,1983
[3]John G.Kemeny,J.Laurie Snell,Finite,Markov Chains,Springer-Verlag,New York Berlin Heidelberg Tokyo,1976
[4]D.Isaacson,R.Madsen,Markov chains theorys and application[M],1976,Wiley,New York
[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][美]钟开莱 概率论教程[M],刘文等译,上海科学技术出版社,1989
[7]朱成熹,随机极限引论[M],天津:南开大学出版社,1987
[8]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
[9]Rosenbllat R M.Some theorems concerning the strong law of numbers for non-homogeneous Markov chains.Ann Math Statist,1964,35;566-576
[10]Liu Guoxin,Liu Wen.On the strong law of large numbers for functionals of countable non-homogeneous Markov chains.Stochastic Processes Appl,1994,50:375-391
[11]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
[12]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
[13]杨卫国,刘开弟,董卫 关于可列非齐次马氏链Cesaro平均收敛性及二元函数的强大数定律,数学物理学报,1998,18(增刊),27-34
[14]C.E.Shannon.A mathematical theory of communication.Bell Syst.Tech.J.,27:379-423,623-656,1948.
[15]B.McMillan.The basic theorems of information theory.Ann.Math.Stat.,24:196-219,1953.[16]L.Breiman.The individual ergodic theorems of information theory.Ann.Math.Stat.,28:809-811,1957.With correction made in 31:809-810.
[17]S K.L.Chung.A note on the ergodic theorem of information theory.Ann.Math.Stat.,32:612-614,1961.
[18].C.Moy.Generalization of the Shonnon-McMillan theorem.Pacific J.Math,pages 705-714,1961.
[19]J.R.Pierce.The early days of information theory.IEEE Trans.Inf.Theory,pages 3-8,Jan.1973.
[20]J.C.Kieffer.A simple proof of the Moy-Perez.generalization of the Shannon-McMillan theorem.Pacific J.Math.,51:203-206,1974.
[21]A.R.Barron.The strong ergodic theorem for densities:generalized Shannon- McMillan-
Breiman theorem.Ann.Prob.,13:1292-1303,1985.
[22]S.Orey.On the Shannon-Perez-Moy theorem.Contemp.Math.,41:319-327,1985.
[23]P.Algoet and T.M.Cover.A sandwich proof of the Shannon- McMillan- Breiman theorem.Ann.Prob.,16(2):899-909,1988.
[24]杨卫国,非齐次马氏链熵率存在定理[J].数学的实践与认识,1993,No.2:86-90.
[25]刘文,杨卫国.相对熵密度与任意二进信源的若干极限性质.应用数学学报.Vol.17 No.11994.
[26]刘文,杨卫国.关于有限马氏链相对熵密度和随机条件熵的一类极限定理,应用数学学报,1995,Vol.18,No.2(234-247)
[27]W.,Liu,W.C.,Yang,A limit theorem for the entropy density of nonhomogeneous Markov information source[J].Statistics & Probability Letters,1995,22:295-301
[28]W.G.,Yang,W.,Liu,The asymptotic equipartition property for Mth-order nonhomogeneous Markov information sources[J].IEEE TRANSANCTIONS ON INFORMATION TEORY,2004,Vol.50,NO.12:3326-3330
[29]刘文,杨卫国.关于非齐次马氏信源的渐进均匀分割性.应用概率统计.Vol.13 No.41997
[30]Yang Weiguo,Liu Wen.Strong law of large numbers and Shannon-McMillan theorem for Markov chains field on Cayley tree[J],Acta Mathmatica Scientia,2001,21B(4):495-502
[31]刘文,王丽英,杨卫国.二进树上奇偶马氏链场的若干强极限定理与Shannon-McMillan定理的一种逼近,应用概率统计,Vol.18,277-285,2002
[32]杨卫国,吴小太,王豹.一类隐马尔可夫模型的若干极限性质[J].江苏大学学报:自然科学版,2006,27(5):467-468.
[33]吴小求.证券投资学[M].北京:中国人民大学出版社.2004.310-321
[34]汪嘉冈。现代概率论基础[M]。上海:复旦大学出版社,1988。
[35]L.,Cover,J.,Thomas,Elements of Information Theory[M].Wiley,New York,1991
[36]Donald R.Van Deventer,Kenji lmia,Mark Mesler.高级金融风险管理[M]译者朱世武,邢丽.中国人民大学出版社,2006
[37]段文斌,桂佳.外汇市场与外汇业务[M].经济管理出版社,2001年
[38]彭志行,夏乐天.马尔科夫链及其在股市分析中的应,应用数学,2004(17)增159—163期
[39]张静,江寿阳.人民币均衡汇率与中国外贸[M].高等教育出版社,2005
[40]Yang Weiguo Strong limit theorems for arbitrary.J.Math.Anal.Appl.326(2007)1445-1451