A Class Of Small Deviation Theorems For Functionals Of Random Fields On Homogeneous Trees

Random fields on trees are applications on trees of theory of stochastic process-a new math model,which developed from coding and encoding problem in information theory.Assuming there is a process {X_t,t∈T},whether the appearing frequency of state and state couple obey the strong law of large numbers is the key of a good coding and encoding method,so this domain is always be a researching emphases for many scholars.Thirty years ago,when random fields came into being, it's a subject of intersection of Probability and Statistic Physics.Random fields, together with other branches of probabilistic Physics,stand for and important aspect of 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.Benjamini and Peres have given the notion of the tree-indexed homogeneous Markov chains and studied the recurrence and ray-recurrence for them(see[1]).Berger and Ye have studied the existence of entropy rate for some stationary random fields on a homogeneous tree(see[2]).Ye and Berger have studied the asymptotic equipartition property(AEP) in the sense of convergence in probability for a PPG invariant and ergodic random field on a homogeneous tree(see[37]).Recently,Yang have studied some strong limit theorems for countable homogeneous Markov chains indexed by a homogeneous tree and the strong law of large numbers and the asymptotic equipartition property(AEP) for finite homogeneous Markov chains indexed by a homogeneous tree(see[26]).Yang and Ye have studied strong theorems for countable nonhomogeneous Markov chains indexed by a homogeneous tree and the strong law of large numbers and the asymptotic equipartition property(AEP) for finite nonhomogeneous Markov chains indexed by a homogeneous tree(see[27]).Liu and Wang have studied the small deviation between the arbitrary random fields and the Markov chain fields on Cayley tree(see[8]). In this paper,by introducing the asymptotic logarithmic likelihood ratio as a measure of Markov approximation of the arbitrary random field on a homogeneous tree,and constructing a non-negative martingale,we first establish a class of small deviation theorems for functional of random fields,then we obtain the strong law of large numbers for the frequencies of occurrence of states and ordered couple of states for random fields,and the asymptotic equipartition property(AEP) for random fields on a Cayley tree.As corollary,we obtain the strong law of large numbers and the AEP for Markov chains indexed by a Cayley tree.In fact,our present outcomes can imply the case in[8]and[26].