A Class Of Limit Theorems For Nonhomogeneous Markov Chains And Tree-indexed Markov Chains

Markov process is an important branch of stochastic processes.In this paper,we mainly study some limit theorems of nonhomogeneous Markov chains,tree-indexed Markov chains and Markov chains in random environment.In the first chapter of this paper,the research background and status of Markov chains,Markov chains in random environment and tree-indexed Markov chains are introduced.Then the chapter arrangement of this paper is introduced.The second chapter is the basic knowledge of this paper,which introduces the concepts,properties and known theorems related to conditional expectation,martingale,Markov chains,tree-indexed Markov chains and Markov chains in random environment.In the first part of the third chapter,we discuss a class of strong deviation theorems of harmonic mean of stochastic transition probability for arbitrary sequences of random variables associated with nonhomogeneous Markov chains.As a corollary,a strong limit theorem for harmonic mean of the random transition probability of nonhomogeneous Markov chains is obtained.In the second part of this chapter,we explore a strong deviation theorem on harmonic mean of stochastic transition probability for arbitrary tree-indexed stochastic process associated with nonhomogeneous tree-indexed Markov chains.As a corollary,the strong limit theorem in [37] on harmonic mean of the stochastic transition probability of nonhomogeneous tree-indexed Markov chains is derived.In the fourth chapter,we mainly study the strong limit theorems for Markov chains in biinfinite random environment.We firstly construct a nonnegative martingle to prove the he strong limit theorems for Markov chains in bi-infinite random environment,and give the strong limit theorem for nonhomogeneous Markov chains.Finally,we obtain the strong limit theorem of harmonic mean of stochastic probabilities for Markov chains in bi-infinite Markov chains.The fifth chapter is the focus of this paper.We study the tree-indexed Markov chain in continuous state.We firstly define the tree-indexed Markov chain in continuous state,and explore the equivalent properties of tree-indexed Markov chain in continuous state.Finally,we prove the strong law of numbers for tree-indexed Markov chains in continuous state.In the sixth chapter,we summarize the main results of this paper and point out the direction for future research.