Some Strong Limit Theorems For Models Related To Markov Chains In Random Environments

Markov chain was first proposed by A.A.Mapkob in the 20 th century,and is used in various disciplines widely because of its unique Markov character.The study of Markov chains in random environments have a long history,and many scholars at home and abroad have made achievements in this field.In the 1970 s,Liu wen,a domestic scholar,established an approach to study the strong limit theorem and proposed the strong deviation theorem creatively.In recent years,Markov chain has been studied in more and more fields,from chain model to tree model,from finite state space to countable state space,the passion for Markov chain has never waned.This paper mainly considers two problems.Firstly,we study the strong law of large numbers of Markov chains in single-infinite Markovian environment on countable state space.,as corollary,we obtain the strong laws of large numbers for the frequencies of occurrence of states and ordered couples of states for this process,meanwhile,we give the asymptotic equipartition property of Markov chains in single-infinite Markovian environment on countable state space.Meanwhile,we considered the the Markov chain indexed by tree in Markovian environment,which based on the conclusion of Yang and Shi's study on Markov chains indexed by tree in random environment,we also proved the equivalence between tree-indexed Markov chains in Markov environment and tree-indexed Markov double chains,and the strong limit property of the harmonic mean of the random transition probability of tree indexed Markov chain in the Markovian environment is obtained under the finite stat a limit property of random transition probability for a Markov chain indexed by a tree in Markovian environment is proved by martingale method.