【摘要】：20世紀(jì)70年代末,劉文教授及其合作者將概率論中的強(qiáng)極限定理推廣到用不等式表達(dá)的情形,建立了隨機(jī)序列的強(qiáng)偏差定理,并取得了豐富的研究成果。本文基于前人的研究基礎(chǔ)上,考慮在不同的參考乘積分布的情形下,研究一類相依隨機(jī)序列關(guān)于不同參考乘積分布的滑動平均的強(qiáng)偏差定理。研究的基本思路是引入滑動似然比和滑動相對熵作為相依隨機(jī)序列聯(lián)合分布與參考乘積分布偏差的一種隨機(jī)性度量。通過限制滑動相對熵的取值范圍,給出樣本空間的一個子集,并在此子集上得到了隨機(jī)序列滑動平均的上、下界,即強(qiáng)偏差定理。證明的要點(diǎn)是構(gòu)造一個帶參數(shù)的滑動似然比,利用經(jīng)典的Borel-Cantelli引理及分析法,得到幾乎處處收斂的不等式。全文共分為六章:第一章是緒論部分,介紹了本論文的選題背景及研究相依隨機(jī)序列強(qiáng)偏差定理的基本思想與方法;第二章簡要介紹了相關(guān)的基本定理和概念,以及與本文有關(guān)的研究成果;第三章引入滑動似然比和滑動相對熵的概念,研究隨機(jī)受控的隨機(jī)序列滑動平均的若干強(qiáng)偏差定理,且推廣已有的成果;第四章繼續(xù)研究連續(xù)信源相對于無記憶Gamma信源的強(qiáng)偏差定理;第五章研究任意相依隨機(jī)序列與可列二重非齊次馬氏泛涵的滑動平均的一類強(qiáng)偏差定理;第六章對全文進(jìn)行總結(jié)與展望。
[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.
【學(xué)位授予單位】：安徽工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O211.4

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