The Strong Limit Theories Of AQSI Sequence

Probability limit theory is one of the important branches and also is an essential theoretical foundation of science of probability and statistics. The famous probability scholar Kolmogorov from previous Soviet Union said:"Only probability limit theory can reveal the epistemological value of probability. Without it,you couldn't understand the real meaning of the fundamental conceptions in probability."Classical limit theory is a significant achievement in the progress of probability. Strong convergence has become the most important and popular direction in the current study of probability limit theory. Some significant results have been reached through deep research in this dissertation.The dependence of random variables as a concept is developed not only in some Branches of Probability theory and mathematical statistics,such as Markov Chains,random Field theory and time series analysis, etc, but also appears in many practical problems. Although the assumption of independence is reasonable sometimes,it is difficult to check The independence of a sample. Moreover,in many practical problems, the samples are not independent observations. Hence one can see that,the study on dependent random variables has momentous significance. The classical limit theorems of probability theory for a sequence of mixing dependent random variables was discussed systematically in Lin and Lu's monograph"Limit Theory on Mixing Denpendent Random Variables"(1997).The definition of asymptotically quadrant sub-independent(AQSI) sequences was introduced by Chandra and Ghosal(1996), This is a class of more extensive sequences, independent ,Pairwise NQD, AQI , As well as many mixed random variables sequences are the special cases of it. Some significant results of strong limit theory for AQSI sequences have been reached through deep research in this dissertation.This article provides five chapters. The first chapter presents an insightful discussion about almost sure convergence and a strong law of large number theorem. As we all know, Kolmogorov type inequality is a useful tool for prove the strong law of large numbers. This chapter shows the Kolmogorov-type inequality of AQSI sequence under some conditions, and then discussed three series theorem and Chung type strong limit theorems. The second chapter discusses the convergence property of Jamison weighted sums.In the third chapter, we discussed the estimate of partial sums for AQSI sequences, through the sum of moment gave the best estimate for S n,and gave the necessary and sufficient condition for AQSI sequences that obey the Kolmogorov strong law of large numbers. The fourth Chapter discusses the the central limit theorem of AQSI sequence. And give some sufficient condition that the central limit theorem found.In the fifth chapter, we discussed the complete convergence for AQSI sequence, and generalized the results of negatively associated sequences under some conditions.