A Class Of Strong Deviation Theorems For Dependent Stochastic Sequence On The Moving Average

Probability theory is a widely used discipline and has close relationship with otherfields. It is the framework foundations of many applied subjects, such as Information,Mathematical Finance and Insurance of Actuaries etc. It is well known that the limittheorem for partial sums of random variables is always one of central problem ofstudying probability. While the probability limit theory for independent randomvariables has been developed well. But in many practical problems, theseresults cannot satisfy the need of reality, so the theory of weakly dependent randomvariables had been widely studied since the nineteen fifties.The purpose of this thesis is to study some strong limit properties for stochasticsequence on moving average. The basic idea is to introduce the notion of the movinglikelihood ratio and the asymptotic logarithmic moving likelihood ratio, as a randommeasure of the deviation between the joint distribution of the arbitrarily dependentrandom sequences and the product of the reference, we give a subset of the samplespace by restricting the asymptotic logarithmic moving likelihood ratio, and obtainlower and upper bounds of the moving average of the partial sums of randomsequences on the subset, to obtain the strong deviation theorem. The main method ofthe proof is to construct a moving likelihood ratio with a parameter and then obtainthe almost everywhere convergence by analytical method.This article is divided into five chapters: in chapter1, we first introduce thebackground of limit theory of probability and development present situation briefly,then give the basic ideas and methods of studying strong limit theorems for dependentrandom sequences in this paper. In the second chapter, we introduce the basicknowledge of probability theory briefly in the article. In the third chapter, we firstintroduce the concept of the run, then discuss some strong limit theorems for runsof dependent sequences of binary random variables for the moving average. In thefourth chapter, we extend the notion of discrete run of the continuous case, and thenobtain some strong deviation theorems of random sequence length of runs to thecontinuous-valued on the moving average. In the fifth chapter, we discuss some strongdeviation theorems for dependent stochastic sequence for the moving average. Theseresults are generalizations of strong limit theorems, which make the resultsmore general, and established a deep connection between the limit theory of probability and information entropy theory. Finally, we summarize this article andpredict the development and application of limit distribution theory of runs.