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Báo cáo hóa học: " Research Article Some Limit Properties of Random Transition Probability for Second-Order"

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 Limit Properties of Random Transition Probability for Second-Order | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 503203 10 pages doi 10.1155 2009 503203 Research Article Some Limit Properties of Random Transition Probability for Second-Order Nonhomogeneous Markov Chains Indexed by a Tree Zhiyan Shi and Weiguo Yang Faculty of Science Jiangsu University Zhenjiang 212013 China Correspondence should be addressed to Zhiyan Shi shizhiyan1984@126.com Received 1 September 2009 Accepted 24 November 2009 Recommended by Andrei Volodin We study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain and a nonhomogeneous Markov chain indexed by a tree. As corollary we obtain the property of the harmonic mean of random transition probability for a nonhomogeneous Markov chain. Copyright 2009 Z. Shi and W. Yang. 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. 1. Introduction A tree is a graph G T E which is connected and contains no circuits. Given any two vertices Ơ t Ơ ft e T let Ơt be the unique path connecting Ơ and t. Define the graph distance d ơ t to be the number of edges contained in the path Ơt. Let To be an arbitrary infinite tree that is partially finite i.e. it has infinite vertices and each vertex connects with finite vertices and has a root o. Meanwhile we consider another kind of double root tree T that is it is formed with the root o of To connecting with an arbitrary point denoted by the root -1. For a better explanation of the double root tree T we take Cayley tree Tc N for example. It is a special case of the tree To the root o of Cayley tree has N neighbors and all the other vertices of it have N 1 neighbors each. The double root tree TC N see Figure 1 is formed with root o of tree TC N connecting with another root -1. Let Ơ t be vertices of the double