Some Shannon-McMillan Approximation Theorems for Markov Chain Field on the Generalized Bethe Tree

Abstract

A class of small-deviation theorems for the relative entropy densities of arbitrary random field on the generalized Bethe tree are discussed by comparing the arbitrary measure μ with the Markov measure μ Q on the generalized Bethe tree. As corollaries, some Shannon-Mcmillan theorems for the arbitrary random field on the generalized Bethe tree, Markov chain field on the generalized Bethe tree are obtained.

Strong Law of Large Numbers for Markov Chains Indexed by Spherically Symmetric Trees

“…Liu Wen and Yang Weiguo [5] studied the deviation theorems for Markov chains field on a generalized Bethe tree or a generalized Cayley tree, in fact the tree model is a special case of uniformly bounded degree tree. Wang Kangkang and Zong Decai [7] tried to establish some Shannon-McMillan approximation theorems for Markov chain field on the generalized Bethe tree, but the results seem crude. In this paper, we drop the uniformly bounded restriction.…”

Some Small Deviation Theorems for Arbitrary Random Fields With Respect to Binomial Distributions Indexed by an Infinite Tree on Generalized Random Selection Systems

“…Since µ and µ Q are two probability measures, it is easy to see that {U n (λ, ω), F n , n ≥ 1} is a nonnegative martingale according to Doob's martingale convergence theorem(see [12]). Hence, we have…”

“…Recently, Yang have studied some limit theorems for countable homogeneous Markov chains indexed by a homogeneous tree and strong law of large numbers and the asymptotic equipartition property (AEP) for finite homogeneous Markov chains indexed by a homogeneous tree (see [7] and [11]). Wang has also studied some Shannon-McMillan approximation theorems for arbitrary random field on the generalized Bethe tree (see [12]). Zhong and Yang (see [14]) have studied some asymptotic equipartition properties (AEP) for asymptotic circular Markov chains.…”