Some Limit Properties For High Order Markov Chains Indexed By A Tree

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.