A Class Of Strong Deviation Theorem For Models Related To Markov Chains In Random Environments

As one of the central problems in probability theory,strong limit theorem is still attractive to many scholars.Strong deviation theorem is the extension of strong limit theorem,which is a kind of strong limit theorem expressed by inequality.The Markov chains in random environment are the Markov chains under the influence of random environment,which make up the limitation of classical Markov chains.In recent years,scholars at home and abroad have made a lot of research achievements in classical Markov model,mainly involving central limit theorem,the strong law of large numbers,Shannon-Mc Millan theorem and other limit theories.Compared with the research results of classical Markov model,the research of Markov model in random environment still has obvious deficiency.In this paper,we generalize some results of Markov chains in discrete state to random environment.We mainly study the tree-indexed Markov model in random environment and the Markov model in single-infinite in random environment.In this paper,the asymptotic likelihood ratio is used as a measure,and the strong deviation theorem of the listed homogeneous Markov chains is established by constructing the method of non-negative martingale.Finally,the strong law of large numbers and the asymptotic equipartition of the listed homogeneous Markov chains are obtained.At the same time,a class of strong deviation theorems for random field functional of double Cayley tree in random environment is established.Finally,the asymptotic equipartition property(AEP)of tree-indexed Markov chains in Markov environment in finite state space is proved.