A Class Of Strong Deviation Theorems For Markov Chain Field On A Cayley Tree

Random fields on trees are applications on trees of theory of random process-a new math model, which developed from coding and encoding problem in information theory.Assuming there is a sequence of {X_n}, whether the appearing frequency of state and statecouple obey the strong law of large numbers is the key of a good coding and encoding method, so this domain is always being a researching emphases for many scholars. Thirty years ago, when random fields came into being, it's a subject of intersection of Probability and Statistical Physics. Random fields, together with other branches of probabilistic Physics, stand for an important aspect of a trend, which is the interpenetration of Math. and Phys.. With the development of the information theory, the tree model has drawn increasing interest from specialist in physics,probability and information theory. Recently professor Yang Weiguo and his associates have studied some limit properties for Markov chains indexed by a homogeneous tree and the asymptotic equipartition property for Even-Odd Markov chain field on a Cayley tree and so on. But the main purpose of this paper is to study the strong deviation theorems for Markov chain field and Even-Odd Markov chain field on a Cayley tree. The strong deviation theorem is a new type of theorems established by introducing the notion of the likelihood ratio. In this paper, we apply the Doob's Martingale convergence theorem and some special inequalities to study. Besides, we obtain some strong deviation theorems for the frequencies of occurrence of states and ordered couples of states for Markov chain field and Even-Odd Markov chain field on a tree.