The asymptotic equipartition property for a special Markov random field

IF 0.8 4区数学 Q3 MATHEMATICS, APPLIED

Stochastics-An International Journal of Probability and Stochastic Processes Pub Date : 2023-09-11 DOI:10.1080/17442508.2023.2255340

Zhiyan Shi, Xiaoyu Zhu

引用次数: 0

Abstract

The asymptotic equipartition property (AEP) plays an important role in establishing lossless source coding theorems and asymptotic coding theorems through the concepts of typical sets and typical sequences in information theory. In this paper, we study the generalized asymptotic equipartition property in the form of moving average for N bifurcating Markov chains indexed by an N-branch Cayley tree, which is a special case of Markov Urandom fields. Firstly, we construct a class of random variables containing a parameter with means of 1, and establish a strong limit theorem for the moving average of multivariate functions of such chains using the Borel–Cantelli lemma. Secondly, we present the strong law of large numbers for the frequencies of occurrence of states of the moving average, as well as the generalized asymptotic equipartition property for N bifurcating Markov chains indexed by an N-branch Cayley tree. As corollaries, we also generalize some known results.

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一类特殊马尔可夫随机场的渐近均分性质

渐近均分性(AEP)在利用信息论中的典型集和典型序列的概念建立无损源编码定理和渐近编码定理方面起着重要的作用。本文研究了以N支Cayley树为索引的N个分岔马尔可夫链的移动平均形式的广义渐近均分性质，这是马尔可夫随机域的一种特殊情况。首先构造了一类参数均值为1的随机变量，利用Borel-Cantelli引理建立了该类链多元函数移动平均的强极限定理。其次，我们给出了移动平均状态出现频率的强大数律，以及以N支Cayley树为索引的N个分岔Markov链的广义渐近均分性质。作为推论，我们也推广了一些已知的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Stochastics-An International Journal of Probability and Stochastic Processes MATHEMATICS, APPLIED-STATISTICS & PROBABILITY

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Stochastics: An International Journal of Probability and Stochastic Processes is a world-leading journal publishing research concerned with stochastic processes and their applications in the modelling, analysis and optimization of stochastic systems, i.e. processes characterized both by temporal or spatial evolution and by the presence of random effects. Articles are published dealing with all aspects of stochastic systems analysis, characterization problems, stochastic modelling and identification, optimization, filtering and control and with related questions in the theory of stochastic processes. The journal also solicits papers dealing with significant applications of stochastic process theory to problems in engineering systems, the physical and life sciences, economics and other areas. Proposals for special issues in cutting-edge areas are welcome and should be directed to the Editor-in-Chief who will review accordingly. In recent years there has been a growing interaction between current research in probability theory and problems in stochastic systems. The objective of Stochastics is to encourage this trend, promoting an awareness of the latest theoretical developments on the one hand and of mathematical problems arising in applications on the other.