Báo cáo hóa học: Research Article Some Shannon-McMillan Approximation Theorems for Markov Chain Field on the Generalized Bethe Tree

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Báo cáo hóa học: " Research Article Some Shannon-McMillan Approximation Theorems for Markov Chain Field on the Generalized Bethe Tree"Hindawi Publishing CorporationJournal of Inequalities and ApplicationsVolume 2011, Article ID 470910, 18 pagesdoi:10.1155/2011/470910Research ArticleSome Shannon-McMillan ApproximationTheorems for Markov Chain Field on theGeneralized Bethe Tree Kangkang Wang1 and Decai Zong2 1 School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang 212003, China 2 College of Computer Science and Engineering, Changshu Institute of Technology, Changshu 215500, China Correspondence should be addressed to Wang Kangkang, wkk.cn@126.com Received 26 September 2010; Accepted 7 January 2011 Academic Editor: Jozef Bana´ s ´ Copyright q 2011 W. Kangkang and D. Zong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 μQ 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.1. Introduction and LemmaLet T be a tree which is infinite, connected and contains no circuits. Given any two verticesx / y ∈ T , there exists a unique path x x1 , x2 , . . . , xm y from x to y with x1 , x2 , . . . , xmdistinct. The distance between x and y is defined to m − 1, the number of edges in the pathconnecting x and y. To index the vertices on T , we first assign a vertex as the “root” and labelit as O. A vertex is said to be on the nth level if the path linking it to the root has n edges. Theroot O is also said to be on the 0th level.Definition 1.1. Let T be a tree with root O, and let {Nn , n ≥ 1} be a sequence of positiveintegers. T is said to be a generalized Bethe tree or a generalized Cayley tree if each vertexon the nth level has Nn 1 branches to the n 1th level. For example, when N1 N 1 ≥ 2and Nn N n ≥ 2 , T is rooted Bethe tree TB,N on which each vertex has N 1 neighboring2 Journal of Inequalities and Applications Level 3 (2,2) (2,5) Level 2 (1,3) (1,1) (1,2) Level 1 (0,1) Root Figure 1: Bethe tree TB,2 .vertices see Figure 1, TB,2 , and when Nn N ≥ 1 n ≥ 1 , T is rooted Cayley tree TC,N onwhich each vertex has N branches to the next level. In the following, we always assume that T is a generalized Bethe tree and denote byT n the subgraph of T containing the vertices from level 0 the root to level n. We use n, j 1 ≤ j ≤ N1 · · · Nn , n ≥ 1 to denote the j th vertex at the nth level and denote by |B| thenumber of vertices in the subgraph B. It is easy to see that, for n ≥ 1, n n n T N0 · · · Nm N1 · · · Nm . 1 1.1 m0 m1 ST , ωLet S {s0 , s1 , s2 , . . .}, Ω ω · ∈ Ω, where ω · is a function defined on T andtaking values in S, and let F be the smallest Borel field containing all cylinder sets in Ω. LetX {Xt , t ∈ T } be the coordinate stochastic process defined on the measurable space Ω, F ;that is, for any ω {ω t , t ∈ T }, define Xt ω ωt, t ∈ T. 1.2 ¸ n n n n XT n μ XT ...