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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2021, Vol. 38 ›› Issue (5): 731-739.doi: 10.3969/j.issn.1005-3085.2021.05.012

• • 上一篇    下一篇

二叉树上非齐次分支马氏链一类强极限定理

李世林,   杨卫国,   石志岩   

  1. 江苏大学理学院，镇江 212013
  • 出版日期:2021-10-15 发布日期:2021-12-15
  • 基金资助:
    国家自然科学基金 (11571142).

Some Limit Theorems for Nonhomogeneous Bifurcating Markov Chains Indexed by a Binary Tree

LI Shilin,   YANG Weiguo,   SHI Zhiyan   

  1. Faculty of Science, Jiangsu University, Zhenjiang 212013
  • Online:2021-10-15 Published:2021-12-15
  • Supported by:
    The National Natural Science Foundation of China (11571142).

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摘要：

近年来树图或者树形网络等诸多复杂系统的结构性质与极限性质逐渐成为研究的热点问题，特别是在树指标马尔可夫链领域的研究中，国内外学者们取得了丰富的研究成果．二叉树上非齐次分支马尔可夫链作为一类特殊的树指标马尔可夫链，该模型的极限性质被国内外学者的广泛研讨并应用于生物动力学、信息论等诸多领域．本文致力于研究在有限状态空间空间取值的二叉树上非齐次分支马尔可夫链转移概率调和平均的极限性质以及该性质与树指标马尔可夫链模型之间的联系．首先在新的条件下，本文给出了在有限状态空间中取值的二叉树上非齐次分支马氏链的强极限定理，并进一步得到了其随机转移概率调和平均的强极限定理，最后借助于两类模型之间的等价关系以及平均值不等式，推广了树指标非齐次马氏链随机转移概率的极限定理．

关键词: 二叉树, 非齐次分支马氏链, 强极限定理, 随机转移概率, 调和平均

Abstract:

Recently, the structural properties and limit properties of many complex systems such as tree graphs or tree networks have become hot topics in research, especially in the field of tree-index Markov chains. A large number of domestic and foreign scholars have obtained numerous significant research results. The non-homogeneous bifurcating Markov chain indexed by a binary tree is a special kind of tree-index Markov chain, and its limit properties have been extensively studied by scholars and applied to many fields such as biodynamics and information theory. This paper is devoted to the limit theorem of the transition probability for non-homogeneous bifurcating Markov chains indexed by a binary tree and the relationship between this theorem and the tree-index Markov chain model. First of all, under new conditions, we present a strong limit theorem for non-homogeneous branch Markov chains indexed by a binary tree taking values in a finite state space. Moreover, the strong limit theorem for the harmonic average of its random transition probabilities has been obtained. Finally, by using the equivalent of the above two models and the mean inequality, we extend the limit theorem of the random transition probability for the non-homogeneous Markov chain indexed by a tree.

Key words: binary tree, nonhomogeneous bifurcating Markov chain, strong limit theorem, random transition probability, harmonic mean

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