The Equivalent Definition Of Second-order Tree-indexed Markov Chain And A Class Of Small Deviation Theorem For The Countable State Homogeneous Markov Chain

Tree-indexed Markov chain is a new mathematical theory system with the mixture of tree and Markov chain. It is a kind of important tree-indexed stochastic processes,which has caused wide-spread concern and inspired a wave of research of mathematicians, biologists, economists, computer workers and so on. We have made significant achievements about its theory research. Xiaoxue Chen, Weiguo Yang and Bao Wang had given the equivalent definition of one-order state. In order to be more convenient to further theoretical research, this paper will generalize previous studies from one-order state to second-order state. All of the work will enrich the content of the theory.At the end of 1980 s, Liu invented the small deviation theorem of the study about probability theory. Later, he studied together with Yang, Chen, Wang and the other people to enrich the small deviation of the theoretical content greatly, and make it to be an independent research branch. Yang also studied the N-valued random variables sequence about m-order non-homogeneous Markov chain and its small deviation theorem. According to the previous research work for the finite Markov chain, this paper will study the small deviation theorem about the countable state homogeneous Markov chain with sequence of random variables. Then, the paper will present the more difficult Shannon-Mc Millan theorem.The research work of this paper are mainly two parts. First part, we will propose the equivalent definition of second-order tree-indexed Markov chain and prove it. All these are in chapter 3. This paper will generalize previous studies from one-order state to second-order state. From this paper, the readers will understand second-order tree-indexed Markov chain more clearly. Then their work of the theory will be better.The second part, we will first prove a class of small deviation theorem for the sequences of countable state random variables with respect to homogeneous Markov chain. Then,we will study the Shannon-McMillan about the sequences of countable state random variables with respect to homogeneous Markov chain. All these are in chapter 4. In the countable state, sum and integral can not exchange positions with each other. So the theoretical method of the finite state is no longer applicable. Now, we break through the advanced finite state theory, and reestablish of relevant strong limit theory. At last, we can solve the the countable state successfully by the use of it and smoothness of conditional probability repeatedly.This paper makes efforts to generalize the results of the predecessors. All the work enriches the content of the limit theory of probability, and makes use of the results of the theory more widely. So, the study of the paper has much theoretical and practical significance.