Several Strong Deviation Theorems For Stochastic Process Indexed By A Tree

The tree-indexed Markov chain is a new research branch in the field of probability theory,which generalizes the Markov chain to the case of tree indicators.As a new theoretical system,it has gained high attention from the field of statistics,information theory,economics,etc.As the basis and important branch of probability theory and mathematical statistics,the strong deviation theorem has been regarded as a hot spot by mathematics researchers.In practice,the transition probability matrix of Markov chain often changes with time,which greatly increases the complexity of the research and attracts more and more people to study the Markov chain on the non-homogeneous tree.This paper is devoted to the study of Markov chain fields indexed by a tree,and gives the concepts of m-ordered Markov chain and double Markov chains on a non-homogeneous tree.In this paper,the stochastic process indexed by a tree is studied by constructing non-negative martingales and combining Doob's martingales convergence theorem with pure analysis method,then several strong deviation theorems and their proofs are given.Firstly,we give a strong limit theorem for m-ordered continuous state non-homogeneous Markov chain indexed by a tree;then we give a class of strong deviation theorems on geometric distribution of Markov chain indexed by a tree;finally,we give a strong limit property for double Markov chains.