Research On Limit Theorems For Nonhomogeneous Markov Chains And Tree-indexed Markov Chains

Probability theory is a mathematical subject that studies the statistical laws for a large number of random phenomena.As one of its major branches,limit theory serves as an important foundation of other branches of probability theory and mathematical statistics.Therefore,it is of great significance to study the limit theory.Markov chain is a kind of special stochastic process,which has become a branch of mathematics with abundant contents.Scholars have done a lot of researches on homogeneous Markov chains,and a complete theoretical system has been formed.However,nonhomogeneous Markov chains still remain an important topic to be further studied.Tree-indexed Markov chain is a new theoretical system generated by the combination of tree graph and Markov chain,which represents an important tree-indexed stochastic process.In recent years,the research of tree indexed-Markov chains has attracted extensive attention of probability theory,computer,physics and other disciplines.Therefore,it has great significance to study tree-indexed Markov chains.In this paper,the limit theorems of nonhomogeneous Markov chains,tree-indexed Markov chains and Markov chains indexed by a Cayley tree in random environment are studied.The research contents are as follows:1.The strong law of large numbers of delayed averages for countable nonhomogeneous Markov chains is studied.Firstly,on the basis of the existing concepts and theorems of the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for nonhomogeneous Markov chains,the application of the generalized C-strong ergodicity for nonhomogeneous Markov chains in information theory is studied.That is,the existence of generalized entropy rate of nonhomogeneous Markov chains under certain conditions is studied.Secondly,it is used Markov Inequality and Borel-Cantelli Lemma to prove the strong limit theoremof bivariate functions for nonhomogeneous Markov chains.Finally,as the research object is countable nonhomogeneous Markov chains,and the operation of countable sum and limit can't be exchanged,the strong law of large numbers of delayed averages of bivariate functions for nonhomogeneous Markov chain is proved by using the smoothness of conditional expectation repeatedly.2.The strong law of large numbers of delayed sums for Markov chains indexed by a Cayley tree in countable state space is studied.Firstly,a strong limit theorem of delayed sums of bivariate functions for Markov chains indexed by a Cayley tree is proved.Then,the strong law of large numbers for the state occurrence frequencies of the delayed sums is obtained.As a corollary,the strong law of large numbers for the state occurrence frequencies of countable Markov chains indexed by a Cayley tree is concluded.3.A class of strong deviation theorems for the random fields associated with bifurcating Markov chains indexed by a binary tree is studied.By introducing the asymptotic logarithmic likelihood ratio as a measure of the deviation between the arbitrary random fields and the bifurcating Markov chains on a binary tree,and constructing a nonnegative martingale,a class of strong deviation theorems for the random fields associated with nonhomogeneous bifurcating Markov chains indexed by a binary tree is derived.And in this paper,the author generalizes the strong law of large numbers(SLLN)and the asymptotic equipartition property(AEP)for the bifurcating Markov chains indexed by a binary tree.4.The generalized entropy ergodic theorem for nonhomogeneous bifurcating Markov chains indexed by a binary tree is studied.Firstly,it is proved a strong limit theorem for delayed sums of the bivariate functions of nonhomogeneous bifurcating Markov chains indexed by a tree.Then,the strong law of large numbers of the frequencies of occurrence of states of delayed sums and the generalized entropy ergodic theorem for nonhomogeneous bifurcating Markov chains indexed by abinary tree are obtained.5.On the basis of the existing definition of tree indexed Markov chains in random environment with countable state space,the realization of Markov chains indexed by a tree in random environment is studied.And the strong law of large numbers and Shannon-McMillan theorem for countable Markov chains indexed by a Cayley tree in a Markovian environment are proved.