Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Some Shannon-McMillan Approximation Theorems for Markov Chain Field on the Generalized Bethe Tree | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 470910 18 pages doi 2011 470910 Research Article Some Shannon-McMillan Approximation Theorems for Markov Chain Field on the Generalized 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 Received 26 September 2010 Accepted 7 January 2011 Academic Editor Jozef Banas Copyright 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 y with the Markov measure yQ 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 Lemma Let T be a tree which is infinite connected and contains no circuits. Given any two vertices x fy e T there exists a unique path x x1 x2 . xm y from x to y with x1 x2 . xm distinct. The distance between x and y is defined to m - 1 the number of edges in the path connecting x and y. To index the vertices on T we first assign a vertex as the root and label it as O. A vertex is said to be on the nth level if the path linking it to the root has n edges. The root O is also said to be on the 0th level. Definition . Let T be a tree with root O and let Nn n 1 be a sequence of positive integers. T is said to be a generalized Bethe tree or a generalized Cayley tree if each vertex on
