Several Strong Deviation Theorems For Markov Chains Indexed By A Non-homogeneous Tree

Probability theory is a subject which studies the random phenomena and reveals laws in quantity.It is a very important branch of applied mathematics and has a wide range of applications in game theory,economics and biology.In1906,Markov studied a special kind of random process in probability theory and named it Markov chains.The study of Markov chains has greatly promoted the application and development of probability theory.Then with the emergence of the tree model and the outstanding performance of the tree model in the fields of probability theory,finance,physics,etc.,a new mathematical system was born,which is extending Markov chains to the tree model.And obtained many meaningful results.As an important branch of probability theory,strong deviation theorems have always been favored by mathematicians.Therefore,it is also extended to the tree model for study,and many excellent conclusions are obtained.The Markov processes that occur in study often results in a variety of states due to different environments and different distributions,which greatly increases the complexity of the study.Therefore,scholars often study the corresponding Markov processes under specific conditions.The main content of this paper is to establish non-homogeneous tree,and construct suitable non-negative martingales based on Markov chains indexed by a non-homogeneous tree.Then some conclusions about for Markov chains indexed by a non-homogeneous tree are given and proved by applying Doob's martingale convergence theorem,the properties of limits and some important inequalities.In Chapter 2,we introduce the concept of generalized gambling system and give a strong deviation theorem for Markov chains indexed by a non-homogeneous tree.In Chapter 3,by combining the generalized geometric distributions,we obtain a strong deviation theorem for the generalized geometric distribution of Markov chains indexed by a non-homogeneous tree.In Chapter 4,we introduce the concepts of double Markov chains,sample divergence and obtain a small deviation theorem for double Markov chains indexed by a non-homogeneous tree.At the same time,the relevant inferences are given and proved,which enriches the content of the strong deviation theorems.