A Class Of Strong Deviation Theorems For The Random Fields Associated With Bifurcating Markov Chains Indexed By A Binary Tree

Markov chain--The original form of the Markov process,was first proposed and named by A.A.Mapkob in the study of stochastic processes in the early 20 th century.This process is widely used in various fields such as modern physics,biology(birth and death process),public utilities and engineering technology because of its Markov characteristics.In recent years,in order to further unlock the unknown fields associated with the Markov chain,many scholars have creatively extended it to the tree model,which has become the focus of people's research.In the late 1970 s,Liu Wen put forward an analytical method for studying the strong limit theorem when studying the strong law of Markov chains.Based on this,a new class of theorem-----strong deviation theorem was established.Benefiting from the research ideas of this method,we mainly study a class of strong deviation theorems for the random fields associated with bifurcating Markov chains indexed by a binary tree in this paper.It is divided into five chapters:Research background and significance,research status at home and abroad,research content and overall layout are given at first.Secondly we give the basic concepts and properties required in this paper,the definition and properties of Markov chains,tree graphs,the definition,properties and some known conclusions of the tree-indexed Markov chains,which will be directly applied in the future and will not be repeated.The third chapter gives a proposition about the integral of random variables is given,which is often applied as a very important known conclusion,but it is not given a clear statement in the relevant literature studied.Therefore,we give this proposition and its proof,and then a conclusion and proof in Chung Kailai's writings are given as a corollary.The fourth chapter is the most important part of the whole paper.By adopting a new method of studying the strong limit theorem of probability theory-----martingale method(which was established by Liu Wen),We study a strong deviation theorem about the random fields with bifurcating Markov chains indexed by a binary tree.Furthermore,the strong law of large numbers and the AEP for the random fields with bifurcating Markov chains indexed by a binary tree are also studied,and the results in reference[5] are generalized.The fifth chapter summarizes the main research results of this paper and reflects on the areas for improvement.