A Class Of Small Deviation Theorems For Random Fields On A Uniformly Bounded Tree

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 [2]. Berger and Ye have studied the existence of entropy rate for some stationary random fields on a homogeneous tree [38]. 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 [1]. 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 [28]. 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 [27]. Huang and Yang [5] have studied strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree. Liu and Wang [9] have studied the small deviation theorems between the arbitrary random fields and the Markov chain fields on Cayley tree. Peng and Yang and Wang [21] have further studied a class of small deviation theorems for functionals of random fields on a homogeneous tree which partially extend the result of [9].In this paper, by introducing the asymptotic logarithmic likelihood ratio as a measure of Markov approximation of the arbitrary random field on a uniformly bounded tree, and by constructing a non-negative martingale, we obtain the following two results:a class of small deviation theorems for functionals of random fields on a uniformly bounded tree and the strong law of large numbers and the asymptotic equipartition property (AEP) for random fields on a uniformly bounded tree. In fact, our present outcomes can imply the case in [5] and [21].