摘要
概率论是从数量上研究随机现象的规律性的学科.它在自然科学、技术科学、管理科学中都有着广泛的应用,因此从上个世纪三十年代以来,发展甚为迅速,而且不断有新的分支学科涌现.概率极限理论就是其主要分支之一,也是概率论的其它分支和数理统计的重要基础.前苏联著名概率论学者Gnedenko和Kolmogrov曾说过：“概率论的认识论的价值只有通过极限定理才能被揭示,没有极限定理就不可能去理解概率论的基本概念的真正含义”.关于独立随机变量的经典的概率极限理论在上世纪30年代和40年代已获得完善的发展,是概率论发展史上的重要成果.二十世纪六十年代以来,继独立随机变量和序列的极限理论获得完善发展之后,各种混合随机变量序列、相伴随机变量序列及鞅的强极限理论又有很大发展,我国学者在这方面做出了许多出色的工作,在国际上也有一定的影响(参见[66,80,84,88,89,118]).信息论的熵定理也称Shannon-McMillan定理或信源的渐进均分割性(AEP),是信息论的基本定理,也是各种编码定理的基础.关于熵定理的最新发展可参考文献[26].
树上的随机场是随机过程理论在树一这一新的数学模型上的应用,它产生于信息理论的编码和译码问题.假设一个序列{Xn，n≥0},其中状态和状态序偶出现的频率是否遵从大数定律,直接影响到编码方法的优劣,故这一领域一直是众多学者研究的重点.三十几年前,诞生的“随机场”这一概率论与统计物理的交叉学科与其它概率物理分支,代表着当今数学与物理相互渗透的大潮流的一个重要侧面.
近年来杨卫国教授与刘文教授合作,采用与传统方法不同的研究方法(参见[37]),在非齐次马氏链强大数定律、信息论熵定理、任意随机变量序列的极限定理、任意离散随机变量序列的强偏差定理及树图上马氏链场的强大数定与熵定理等方面进行了一系列研究,在国内外重要学术刊物上发表了一系列论文[37-65,74-75,81-83,93-112,119].本博士论文在杨卫国教授和刘文教授的研究基础上,进一步研究了树上高阶马氏链的强大数定律和熵定理,以及强偏差定理,推广了杨卫国等研究的结果.
本博士论文共分为七章：
第一章：基本概念,主要结论和方法介绍.
第二章：研究了广义Cayley树上二重马氏链的的强极限理论,作为推论得到了广义Cayley树上二重马氏链状态序偶频率的极限定理,同时也得到了广义Cayley树上二重马氏链强大数定律和Shannon-McMillan定理.
第三章：研究了广义一致有界无穷树上二重马氏链的的强极限理论,作为推论得到了广义一致有界无穷树上二重马氏链状态序偶频率的极限定理.最后,得到了广义一致有界无穷树上二重马氏链强大数定律和Shannon-McMillan定理.
第四章：研究了m根Cayley树上m阶非齐次马氏链的的强极限理论,作为推论得到了m根Cayley树上m阶非齐次马氏链状态序偶频率的极限定理.最后,得到了m根Cayley树上m阶非齐次马氏链在a.e收敛意义下的强大数定律和Shannon-McMillan定理.
第五章：在m根Cayley树上,通过任意测度与m阶非齐次马氏测度比较,研究了m根Cayley树上任意随机场关于m阶非齐次马氏链的强偏差定理,作为推论,得到了m根Cayley树上一类m阶非齐次马氏链的强大数定律与熵定理.
第六章：研究树上路径过程的随机条件概率的调和平均的极限性质.
第七章：研究有限无穷树上二阶非齐次马氏链和非齐次马氏链的随机转移概率的调和平均的极限性质.
Theory of Probability is a science of quantitatively studying regularity of random phenomena,which is extensivelapplied in natural science,techno-logical science,and managerial science etc. Hence, it has been developing rapidly since 1930s and many new branches have emerged from time to time. Limit theory is one of the important branches and also an essential theoretical basis of science of probability and statistics. As stated by Gendenko and Kolmogrov,"The epistemological value of the theory of probability is revealed only by limit theorems. Without limit theorems it is impossible to understand the real content of the primary concept of all our sciences the concept of probability." The classical limit theorems of probability theory for independent random variables had been developed successfully in 1930 s and 1940 s,and they are the significant achievements in the progress of probability. In 1960's, the limit theorem for the sequences of indepent random variables has been well established. Since then, the limit theorem for mixing sequences of random variables and correlated sequence of random variables has been greatly developed. Many Chinese researchers have contributed outstandingly in this field. Their influential works have been international recognized (Cf[66,80,84,88,89, 118]). The entropy theorem in information theory, which is of core interst in this thesis, is also frequently as the Shannon-McMillan theorem or asymp-totic equipartition property(AEP).It is fundamental theorem in information theory which lays the foundamental theorem in informatition theory, which also lays the foundatition of almost all the coding theorems. The most recently development of entropy theorem could be found in [26].
Random fields on trees are applications on tree of theory of stochastic process—A new math model, which developed from coding and encoding problem in information theory. Assuming there is a sequence of{Xn}, whether the appearing frequency of state and state couple obey the strong law of large numbers is a key of a good coding and encoding method, so this domain is always being a researching emphases for many scholars thirty years ago, when random field came into being. It is a subject of intersection of probability and statistical physics. Random field, together with other branches of probabilistic physics, stand for an important aspect of a trend, which is the interpenetration of Math and Phys.
In recent years, Yang and Liu have studied the strong law of large numbers for nonhomogeneous Markov chains, entropy theorems in information theory, strong limit theorem for arbitrary stochastic sequences, strong deviation theorem for sequences of discrete random variables and strong law of large numbers and entropy theorem for Markov chains fields on trees by using the new approaches which are different from traditional ones. Many papers have been published in the national journal and international journals (Cf [37-65,74-75,81-83,93-112,119]). Many results can also been found in book [41]. This doctoral dissertation based on Yang and Liu's research, further study the strong law of large numbers、entropy theorems、strong deviation theorem of high Markov chains indexed by tree, and extends Yang,etc's results.
There are seven chapters in this doctoral dissertation.
In chapter 1, we give an introduction of the basic notations, main results and approches used in this paper.
In chapter 2, we first study a local convergence theorem for a finite second order Markov chain indexed by a general Cayley tree. As corollaries, we obtain some limit theorems for this Markov chain. Finally, we obtain the strong law of large numbers(LLN) and Shannon-McMillan theorem for a class of finite second order Markov chain indexed by a general Cayley tree.
In chapter 3, we study a local convergence theorem for a finite second order Markov chain indexed by a general infinite tree with uniformly bounded degree. As corollaries, we obtain some limit theorems for this Markov chain. Finally, we obtain the strong law of large numbers (LLN) and Shannon-McMillan theorem for a class of finite second order Markov chain indexed by a general infinite tree with uniformly bounded degree.
In chaper 4, we first study a convergence theorem for a finite m th-order nonhomogeneous Markov chain indexed by an m rooted Cayley tree. As corollaries, we obtain some limit theorems for the frequencies of occurrence of states for this Markov chain. Finally, we obtain the strong law of large numbers and Shannon-McMillan theorem for a class of finite m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree.
In chapte 5, we are to establish a class of strong deviation theorems for the random fields relative to m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree. As corollaries, we obtain the strong law of large numbers and Shannon-McMillan theorem for m th-order nonhomogeneous Markov chains indexed by that tree.
In chapter 6, we study a limit property of the harmonic mean of random path conditional probability for path process indexed by a tree. As corollary, we obtain the properties of the harmonic mean of random conditional probability of a sequence of random variables and a nonhomogeneous Markov chain indexed by a tree.
In chapter 7, we study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain and a nonhomogeneous Markov chain indexed by a tree. As corollary, we obtain the property of the harmonic mean of random transition probability for a nonhomogeneous Markov chain.
引文
[1]Ash, R.B., Probability and MeasureTheory,2nd Edition, Academic Press,Burlington, 2000.
[2]AJgoet,P.H.and CoverT.M.,A sandwich proof of the Shannon-McMillan-Breiman theorem, Ann. Probab.,16,899-909,1988.
[3]Barron, A.R., The strong ergodic theorem for densities:generalized Shannon-McMillan Breiman theorem, Ann.Probab.,13,1292-1303,1985.
[4]Benjamini, I., Haggstrom, O., Peres, Y., and Steif,J.E., Which properties of a random sequence are dynamically sensitive? Ann.Probab.,31,1-34,2003.
[5]Benjamini,I.and Peres,Y., Markov chains indexed by trees, Ann.Probab.,22,219-243,1994.
|6] Benjamini, I., and Peres,Y, Tree-indexed random walks on groups and First passage Percolation.Probab. Theory Related fields,98,911-1121,1992.
[7]Breiman,L,The individual ergodic theorem of information theory, Ann,Math.Statis.,28, 809-811,1957.
[8]Bowerman,B., David, H.T.and Isaacson,D., The convergence of Cesaro averages for certain nonstationary Markov chains,Stochastic Process. Appl,5,221-230,1977.
[9]Berger,T and Ye, Z., Entropy aspects of random fields on treess, IEEE Trans. Inform. Theory.36,1006-1018,1990.
[10]Blum, J.R., Hanson, D.L., Koopmans, I., On the strong law of large numbers for a class of stochastic Processes, Z.Wahrschein, Verse. Geb,2,1-11,1963.
[11]Cantrell, A.and Rosalsky, A., Some strong laws of large numbers for Banach space-valued summands respective of their joint distributions, Stochastic Anal.Appl,21,79-95,2003.
[12]Chen, M. F.and Wang, Y.Z., Algebraic convergence of Markov chains, Ann.Appl.Probab.,13, 604-627,2003.
[13]Chow,Y.S., Teieher,H., Probability Thoery,2nd Edition, Springer, New York,1988.
[14]Chung, K.L., A Course in Probability Theory,2nd Edition, Academic Press, New York,1974.
[15]Chung, K.L., Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York,1967.
[16]Chung, K.L., The ergodic theorem of information theory. Ann. Math. Statist.32,612-614, 1961.
[17]Champagnat, E, Idier, J.and Gousard, Y, Stationary Makov random fields on a finite rectangular lattice, IEEE. Trans.Inform. Theory,44,2901-2916,1998.
[18]Chung, K.L., The ergodic theorem of information theorem, Ann.Math.Statist.,32, 612-614,1961.
[19]Cover, T. M and Thomas, J.A., Elements of Information theory, Wiley, New York,1991.
[20]Doob, J.L.,Stochastic Proeesses,Wiley,New York,1953.
[21]Dubins, L.E.and Freedman, D.A., A sharper form of the Borel-Cantelli lemma and strong law,Ann.Math.Statist.,36,800-807,1965.
[22]Freedman, D., A remark on the strong law, Ann.Probab.,2,324-327,1974.
[23]Freedman, D., Another note on the Borel-Cantelli Lemma and the strong law with the possion Approximation as a by-Product, Annals of Probability,1,910-925,1973.
[24]Follmer, H. On entropy and information gain in random fields, Z,Wahrscherin, Verse, Geb 26,207-217,1973.
[25]Georgii, H.O.,Gibbs Measures and Phase Transitions, Walter de Grugter,Berlin,1988.
[26]Gray R.M., Entropy and Information Theory. Springer-Verlag, New York,1990.
[27]Huang, H.L., Yang, W.G., Strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree. Sci. China Ser. A 51(2),195-202,2008.
[28]Hull, J.C., Options, Futures, and Other Derivative Securities,3rd Ed., Prentice.Hall,1993.
[29]Hall,P., Heyde, C.C., Martingale Limit Theory and its Application. Academic Press, New York,1980.
[30]Isaac,R.,On equitable ratios of Dubins-Freedman type, Statist Probab. Lett,42,1-6,1999.
[31]Isaaeson, D.L.,and Madsen, R.W., Markov Chains Theory and Applications, Willey, New York,1979.
[32]Jajte, R., On the strong law of larger numbers,Ann.Probab,31,409-412,2003.
[33]Jardas, C and Pecarie, J., A note on Chung's strong law of large numbers, J.Math Analysis. Appl,217,328-334,1988.
[34]Jarner,S.F. and Roberts,G.O., Polynomial convergence rates of Markov chains., Ann,Appl. Probab,12,224-247,2002.
[35]Kemeny J.G.,Snell,J.L.and Knapp,A.W, Denumerable Markov Chains, Springer-Verlag, New York,1976.
[36]Karatzas,I.and Shreve, S.E. Methods of Mathematieal Finanee, Springer-Verlag, New York,1998.
[37]Liu, W. Relative entropy densities and a class of limit theorems of the sequence of Valued random variables, Ann.Probab.,18,829-879,1990.
[38]Liu, W., Some limit properties of the multivariate function sequences of diserete random Variables, Statistic&Probability Letter,61,41-50,2003.
[39]Liu.W., Wang. L.Y., The Markov approximation of the random field on Cayley tree and a class of small deviation theorems, Statistic&Probability Letter.63,113-121,2003.
[40]刘文,一个对离散信源普遍成立的强极限定理,数学物理学报,16,440-443,1996.
[41]刘文,强偏差定理与分析方法,科学出版社,2003.
[42]刘文,陈爽,杨卫国,关于样本熵的一类强偏差定理,应用数学学报,23,610-619,2000.
[43]Liu,W, A limit property of random conditional probabilities, Statistic&Probability Letter, 49:299-304,2000.
[44]刘文,随机条件概率的一个极限性质与条件矩母函数方法.应用数学学报,2000,23(2)：275-279.
[45]刘文,有限非齐次马氏链随机转移概率调和平均的一个强极限定理.数学物理学报,2000,20(1)：81-84.
[46]刘文,杨卫国,一类对可列非齐次马氏链普遍成立的强大数定律,科学通报,37,1448-1451,1992.
[47]刘文,杨卫国,任意信源与马氏信源比较及小偏差定理,数学学报,40,22-36,1997.
[48]刘文,杨卫国,关于任意随机变量序列的一类强极限定理,数学学报,40,537-543,1997.
[49]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,15,479-489,1992.
[50]刘文,杨卫国,相对熵密度与任意二进信源的若干极限定理,应用数学学报,17,85-99,1994.
[51]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,18,234-247,1995.
[52]刘文,杨卫国,树上马氏链场的若干极限性质,数学物理学报,21,512-520,2001.
[53]刘文,杨卫国,关于Shannon-McMillan定理的若干研究，数学物理学报,14,337-345,1994.
[54]刘文,杨卫国,关于Shannon-McMillan定理的若干研究(2),数学物理学报,17,176-183.1997.
[55]刘文,杨卫国,可列非齐次马氏链的一类极限定理,应用概率统计,8,64-69,1992.
[56]刘文,杨卫国,任意信源二元函数一类平均值的极限性质,应用概率统计,11,195-203.1995.
[57]刘文,杨卫国,非齐次马氏信源的渐近均分割性,应用概率统计,13,32,59-366,1997.
[58]Liu. W. and Yang, W.G., An extension of Shannon-McMillan theorem and some limit Properties for nonhomogeneous Markov chains, Stochastic Process.Appl.,61,129-145,1996.
[59]Liu, W. and Yang, W.G., The Markov approximation of the sequences of N-valued random variables and a class of small deviation theorems, Stochastic. Process.Appl,89,117-130,2000
[60]Liu, W. and Yang, W.G., A limit theorem for the entropy density of nonhomogeneouous Markov information source, Statistics&Probability Letter,22,295-301,1995.
[61]Liu, W. and Yang, W.G., A class of strong limit theorems for the sequences of arbitrary random variables, Statistics&Probability Letter,64,121-131,2003.
[62]Liu, W.and Yang, W.G., Some limit properties of nonhomogeneous Markov information source Journal of Combinatrics Information and System Science,21,211-223,1995.
[63]Liu. W. and Yang. W.G., Some extensions of Shannon-McMillan theorem, Journal of Combinatrics Information and System Science,21,211-223,1996.
[64]Liu, W., Yan, J.A. and Yang, W.G,A limit theorem for Partial sum of random variables and its applications, Statistic&Probability Letter,62,79-86,2003.
[65]Liu, W. and Yang, W,G, A strong limit theorem expressed by inequalities for the sequences of absolutely continuous random variables, Hiroshima. Math.J.Japan,32,379-387,2002.
[66]陆传荣,林正炎,相依变量的极限理论,科学出版社,1997.
[67]Loeve, M., Probability Thoery 4th Edition, Springer Berlin, New York,1978.
[68]McMillan, B.The basic theorems of information theory,Ann,Math,Statist,24,196-216,1953.
[69]Meyn,S.P. and Tweedie,R.L.,Markov Chains and Stochastic Stability,Springer:London,1993.
[70]Merton,R.C., Continuous-Time Finance,Oxford,Basil Blackwell,1992..
[71]Ninness,B.,Strong laws of large numbers under weak assumptions with application, IEEE Trans Automat. Control,45,2117-2122,2000.
[72]Petrov, V.V., Sums of Independent Random Variables, Springer, New York,1975.
[73]Pemantle,R., Antomorphism invariant measure on trees, Ann..Probab.20,1549-1566,1992.
[74]Peng W,C, Yang, W,G,and Wang,B, A class of small deviation theorems for functionals of random fields on a homogeneous tree, J. Math. Anal. Appl.361293-301,2010.
[75]彭维才,杨卫国,关于奇次树上随机场的一类强偏差定理,系统科学与数学,29(5),577-584.
[76]Pickard, D.K.,Unilateral Markov fields,Adv, Appl.Probab,12,655-671,1980.
[77]Pliska, S.R,Introduction to Mathematical Finance:Diserete Time Models, Blackwell,1997.
[78]Riehard, S.Ellis, Entrop,Large Deviations and Statistical Mechanics, Springer-Verlag, NewYork,1985.
[79]Shannon,C., A mathematical theory of communication, Bell System Tech..J,27,379-423, 623-656,1948.
[80]邵启满,关于一混合序列的完全收敛性,数学学报,32(1989),377-393.
[81]石志岩,杨卫国,树上非齐次马氏链随机转移概率的极限性质,应用数学学报,31,648-653,2008.
[82]Shi,Z.Y and Yang,W.G, Some Limit Properties of Random Transition Probability for Second-Order Nonhomogeneous Markov Chains Indexed by a Tree, Journal of Inequalities and Applications,ID 503203,2009.
[83]Shi Z.Y and Yang W.G, Some limit properties for the mth-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree, Statistics & Probability Letters,80: 1223-123,2010.
[84]苏淳,王岳宝,同分布NA序列的强收敛性,应用概率统计,14,130-140，1998.
[85]Spitzer,E, Markov random fields on infinte tree. Ann.Probab.,3,387-396,1975.
[86]Shannon, C.E., A mathematical theory of communication. Bell Syst. Tech.,27,379-423, 623-656.,1948.
[87]Takacs, C., Strong law of large numbers for branching Markov chains, Markov Process Related Fields,8,107-116,2002.
[88]于定成,苏淳,冷劲松,NA序列广义Jamison型加权和的几乎处处收敛性,应用数学学报,25,77-87,2002.
[89]王岳宝,刘许国,梁琼,NA列广义Jamison型加权和的强稳定性,科学通报,42,2375-2379,1997.
[90]Winkler, Image Analysis, Random Fields and Dynamic MonteCarlo Methods, Springer-Verlag,New York,1995.
[91]汪嘉冈,现代概率论基础,复旦大学出版社,1988.
[92]严加安,测度论讲义,第二版,科学出版社,2004.
[93]杨卫国等,关于可列非齐次马氏链的Cesaro平均收敛性及二元函数强大数定律,数学物理学报,27-34,1998增刊.
[94]杨卫国等,可列非齐次马氏链二元泛函的强大数定律,应用概率统计,14,381-385,1998.
[95]Yang, W.G., Convergence in the Cesaro sense and strong law of large number for nonhomogeneous Markov Chains, Linear Algebra and Its Applications,354,275-288,2002.
[96]Yang W.GSome limit properties for Markov chains indexed by a homogeneous tree, Statistics & Probability Letter,65,241-250,2003.
[97]Yang, W.G. and Liu, W., Strong law of large numbers for Markov chains field on a Bethe tree, Statistics & Probability Letter,49,245-250,2000.
[98]Yang, W.G., Asympotoic equipartition property for nonhomogeneous Markov information souree, Probability in the Engineering and information Sciences,12,509-518,1998.
[99]Yang, W.G., Strong limit theorems for arbitrary stochastic sequences, J.Math.Anal.Appl,326, 1445-1451,2007.
[100]杨卫国,刘文,关于非齐次m阶马氏信源的渐近均分割性,应用数学学报,25,686-693,2002.
[101]杨卫国,刘文,关于任意随机序列的强收敛性,数学物理学报,23,565-572,2003.
[102]杨卫国,刘文,关于非齐次二重马氏信源的若干极限定理,应用概率统计,15,113-123,1999.
[103]Yang, W.G, and Liu, W., Strong law of large numbers and Shannon-McMillan theorem for Markov Chain fields on trees, IEEE Transaction on Information Theory,48,313-318, 2002.
[104]Yang, W.G.and Liu, W., Strong law of large numbers and Shannon-McMillan theorem for Markov Chain field on Cayley tree, 数学物理学报(英文版),21,495-502,2001.
[105]Yang,W.G.,and Liu, W.,The Asymptotic equipartition property for IEEE Transaction on I n formation Theory,50,3326-3330,2004.
[106]杨卫国,刘文,关于齐次树指标可列马氏链的若干极限性质,工程数学学报,769-773,2004.
[107]Yang, W.G., Ye, Z.and Liu, W., A local convergence theorem for partial sums of stochastic adapted sequences, Czechosloeak Mathematical Journal,56,525-532,2006.
[108]Yang, W.G. and Ye,Z., The Asymptotic Equipartition property for Markov chains indexed by a Homogeneous tree, IEEE Trans Inf Theory,53(9):3275-3280,2007.
[109]杨卫国，叶中行,包振华,关于任意随机适应序列部分和的一类局部极限定理,系统科学与教学,26,187-192,2006.
[110]杨卫国，叶中行,Cayley树图上奇偶马氏链场的渐近均分割性,应用概率统计,22,35-42,2006.
[111]杨卫国,关于强极限定理的若干研究及应用,上海交通大学博十学位论文,2007.
[112]杨卫国,仟意N值随机变量序列关于m阶非齐次马氏链的一类小偏差定理,29A(2)：517-527,2009.
[113]叶中行,林建忠,数理金融,科学出版社,2002.
[114]叶中兴,信息论基础,高等教育出版社,2004.
[115]Ye. Z., and Berger,T., Information Measure for Diserete Random Fields, Science Press, Beijing,1998.
[116]Ye, Z.and Berger, T, Ergodicty regularigy and asymptotic equipartition property of random fields on trees J.Comb Inf. Syst Sci,21,157-184,1996.
[117]雍炯敏,刘道百,数学金融学,上海人民出版社,2003.
[118]张立新,B值强混合随机场的进一步矩不等式及强大数律,应用数学学报,23,518-523,2000.
[119]Zhong,P.P. Yang, W.G,, and Liang,P,P, The Asymptotic Equipartition Property For Asymptotic Circular Markov Chains,Probability in the Engineering and Infor-mational Sciences,24,279-288,2010.