| 摘要: |
| 本文研究了一类定义在二根齐次树上的二阶马尔可夫随机场,其节点状态依赖于前两个祖先节点的分布。通过引入“-1层”根节点,构建了新的联合概率分布模型,推广了马氏链经典Shannon-McMillan定理及小偏差定理。进一步,通过构造样本相对熵率作为偏差度量,建立了针对任意三变量函数极限性质的小偏差定理,揭示了状态转移频率与理论期望的渐近偏差上界。本文的模型为分析复杂层级依赖系统提供了新的理论工具。 |
| 关键词: Shannon-McMillan定理 二根树 任意随机场 二阶马尔可夫随机场 样本相对熵密度。 |
| DOI： |
| 分类号:O211.6 |
| 基金项目:国家自然科学资助(61773012),江苏省“六大人才高峰”工程资助(JY-082), 江苏省“333工程”项目资助 |
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| A Class of Small Deviation Theorems for Multivariate Functions of Second-Order Markov Random Fields Indexed by a Two-Rooted Tree |
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wangxinyu, wangkangkang
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Jiangsu University of Science and Technology
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| Abstract: |
| This paper investigates a class of second-order Markov random fields defined on two-rooted homogeneous trees, where the state of each node depends on the distribution of its two ancestor nodes. By introducing a "-1st layer" root node, a novel joint probability distribution model is constructed, extending the classical Shannon-McMillan theorem and the small deviation theorem for Markov chains. Furthermore, by defining the sample relative entropy rate as a deviation measure, a small deviation theorem is established for the limiting properties of arbitrary three-variable functions, revealing the asymptotic upper bound of deviations between state transition frequencies and their theoretical expectations. The proposed model provides a new theoretical framework for analyzing complex hierarchical dependency systems. |
| Key words: Shannon-McMillan Theorem Two-Rooted Tree Arbitrary Random Field Second-Order Markov Random Field Sample Relative Entropy Density. |
