相依随机序列滑动平均的强偏差定理的研究
发布时间：2019-02-09 16:36
【摘要】：20世纪70年代末,刘文教授及其合作者将概率论中的强极限定理推广到用不等式表达的情形,建立了随机序列的强偏差定理,并取得了丰富的研究成果。本文基于前人的研究基础上,考虑在不同的参考乘积分布的情形下,研究一类相依随机序列关于不同参考乘积分布的滑动平均的强偏差定理。研究的基本思路是引入滑动似然比和滑动相对熵作为相依随机序列联合分布与参考乘积分布偏差的一种随机性度量。通过限制滑动相对熵的取值范围,给出样本空间的一个子集,并在此子集上得到了随机序列滑动平均的上、下界,即强偏差定理。证明的要点是构造一个带参数的滑动似然比,利用经典的Borel-Cantelli引理及分析法,得到几乎处处收敛的不等式。全文共分为六章:第一章是绪论部分,介绍了本论文的选题背景及研究相依随机序列强偏差定理的基本思想与方法;第二章简要介绍了相关的基本定理和概念,以及与本文有关的研究成果;第三章引入滑动似然比和滑动相对熵的概念,研究随机受控的随机序列滑动平均的若干强偏差定理,且推广已有的成果;第四章继续研究连续信源相对于无记忆Gamma信源的强偏差定理;第五章研究任意相依随机序列与可列二重非齐次马氏泛涵的滑动平均的一类强偏差定理;第六章对全文进行总结与展望。
[Abstract]:In the late 1970s, Professor Liu Wen and his collaborators extended the strong limit theorem of probability theory to the case of inequality, established the strong deviation theorem of random sequence, and obtained abundant research results. In this paper, based on the previous studies, we consider the strong deviation theorems for a class of dependent random sequences with respect to the moving average of different reference product distributions in the case of different reference product distributions. The basic idea of the study is to introduce the sliding likelihood ratio and sliding relative entropy as a random measure of the deviation between the joint distribution and the reference product distribution of dependent random sequences. By limiting the range of sliding relative entropy, a subset of the sample space is given, and the upper and lower bounds of the moving average of random sequences, that is, the strong deviation theorem, are obtained. The main point of the proof is to construct a sliding likelihood ratio with parameters. By using the classical Borel-Cantelli Lemma and the analysis method, we obtain almost everywhere convergence inequalities. The full text is divided into six chapters: the first chapter is the introduction part, which introduces the background of this paper and the basic ideas and methods of studying the strong deviation theorem of dependent random sequence; The second chapter briefly introduces the relevant basic theorems and concepts, as well as the research results related to this paper; In chapter 3, the concepts of sliding likelihood ratio and sliding relative entropy are introduced to study some strong deviation theorems for sliding average of random controlled random sequences. In chapter 4, we continue to study the strong deviation theorems of continuous sources relative to memoryless Gamma sources, and in chapter 5, we study a class of strong deviation theorems for the moving mean of random sequences with arbitrary dependent sequences and countable doubly nonhomogeneous Markov universal culverts. Chapter six summarizes and prospects the full text.
【学位授予单位】：安徽工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.4
本文编号：2419157
[Abstract]:In the late 1970s, Professor Liu Wen and his collaborators extended the strong limit theorem of probability theory to the case of inequality, established the strong deviation theorem of random sequence, and obtained abundant research results. In this paper, based on the previous studies, we consider the strong deviation theorems for a class of dependent random sequences with respect to the moving average of different reference product distributions in the case of different reference product distributions. The basic idea of the study is to introduce the sliding likelihood ratio and sliding relative entropy as a random measure of the deviation between the joint distribution and the reference product distribution of dependent random sequences. By limiting the range of sliding relative entropy, a subset of the sample space is given, and the upper and lower bounds of the moving average of random sequences, that is, the strong deviation theorem, are obtained. The main point of the proof is to construct a sliding likelihood ratio with parameters. By using the classical Borel-Cantelli Lemma and the analysis method, we obtain almost everywhere convergence inequalities. The full text is divided into six chapters: the first chapter is the introduction part, which introduces the background of this paper and the basic ideas and methods of studying the strong deviation theorem of dependent random sequence; The second chapter briefly introduces the relevant basic theorems and concepts, as well as the research results related to this paper; In chapter 3, the concepts of sliding likelihood ratio and sliding relative entropy are introduced to study some strong deviation theorems for sliding average of random controlled random sequences. In chapter 4, we continue to study the strong deviation theorems of continuous sources relative to memoryless Gamma sources, and in chapter 5, we study a class of strong deviation theorems for the moving mean of random sequences with arbitrary dependent sequences and countable doubly nonhomogeneous Markov universal culverts. Chapter six summarizes and prospects the full text.
【学位授予单位】：安徽工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.4
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