Some Limit Theorems For Nonhomogeneous Markov Chains Indexed By A Binary Tree And Nonhomogeneous Markov Chains

In recent years,Tree-indexed Markov chain is a new mathematical theory system which is the mixture of tree and Markov chain.Scholars at home and abroad has conducted a lot of related researches on the tree-indexed Markov chain.Dang,Yang and Shi gave the definition of the discrete form of the Markov chain on the binary tree in2015.Meanwhile,they studied the equivalence definition,the strong law of large number and the Shannon-McMillan theorem for nonhomogeneous bifurcating Markov chain indexed by a binary tree.This paper aims to explore the strong law of large number of ternary function for nonhomogeneous bifurcating Markov chain indexed by a binary tree.Chinese researchers Chen and Yang have done much work on the ergodicity for Markov chain.The C-strong ergodicity for Markov chain is a basic and important part of the ergodicity theorems for Markov chain.This paper presents a generalization of the C-strong ergodicity for Markov chain.The author obtains the generalized C-strong ergodicity for the nonhomogeneous Markov chains.Firstly,this paper mainly introduces the research background,and a simple plan of the structure,then it also introduces the concepts,properties and theorems.Then,this paper gives the definition of the random variable set and the random order set of the bifurcating Markov chains indexed by a binary tree.And it concludes the strong law of large numbers for the frequencies of occurrence of the random ordered couples of finite states for bifurcating Markov chain indexed by a binary tree.The strong law of large numbers are studied for functions of the bifurcating Markov chains indexed by a binary tree.As a corollary,the author obtains the Shannon-McMillan theorem for the bifurcating Markov chains indexed by a binary tree with finite state space.Secondly,this paper gives the definition of the generalized C-strong ergodicity and the generalized uniform C-strong ergodicity for countable nonhomogeneous Markov chains,and then it studies the sufficient conditions of the two ergodicity.Finally,the author summarizes the research content of this paper,at the same time,points out the deficiencies that exist in the research process and explains the direction of improvement in the research process.