| Probability limit theory is one of the main branches of probability theory and also an important foundation in other branches of probability theory and mathematical statistics. All along, the independent random variable is the basic objects of the probability limit theory study, and the classical theory of independent random variables have been well thought out development in the 20th century, 30's and 40's. The random variables that come from practical problems are usually not independent. There are always some dependences or another among random variables. Therefore, the properties of the dependent random variables had drawn many attentions from scholars.The introduction of a number of dependent random variables is not only the needs of a theoretical study, such as markov chain, random theory, a branch of multivariate statistical analysis had been made on some random variables in the concept of dependency, but also the needs of practical problems, such as the some practical observations in the statistical sample of the existence of non-independent situation.Negative association random variables (NA) and martingale difference sequences are very important cases in dependent random variables. Negative association random variables were introduced in the statistical literature and had found many application in reliability theory and percolation theory. Martingale difference sequences were the natural extension of independent random variables,which were introduced into modern probabilistic literature by Ville in 1939. The concept of martingale difference is of important significance in theory and application.This thesis consists of four chapters.In chapter 1, we discuss the precise asymptotics in the Baum-Katz law of large numbers for linear processes of martingale difference sequences. Tan[28]proved the precise asymptotics in the Baum-Katz law of large numbers for moving-average processes of positive association sequences. In this chapter, inspired by Tan[28], we consider the precise asymptotics in the Baum-Katz law of large numbers for linear processes of martingale difference sequences. Between positively associated and the martingale difference sequences, there are different in the nature of dependency structure. Therefore, this article of the proof method and the added conditions, lemma are some differences. First, by using moment inequality of martingale difference sequences and some known conditions, moment inequalities of linear process of martingale difference sequences are established. And then by the lemmas of several related, the exact asymptotic behaviors are proved in Baum-Katz law of large numbers for linear processes of martingale difference sequences.In chapter 2, we discuss the precise asymptotics for linear processes of nonstationary martingale difference sequences. there are many nonstationary sequences in pratical prombles, therefore, strong stationary condition are bound to hinder the study about some issues.The main purpose of work is to remove the strong stationary restriction.In the second chapter, we have removed the shackles of stable conditions and further promote the conclusions of the first chapterIn chapter 3, we discuss the precise asymptotics for partial sums of normal attraction nonstationary NA sequences. In literature [4], the auther discussed the precise asymptotics for partial sums of IID and NA sequence. The conclusions of literature [4] were seted up under strong stable conditions. In this chapter, inspired by literature [4], we considers the precise asymptotics for partial sums of normal attraction nonstationary NA sequences, and the results contain some existed conclusions.In chapter 4, we discuss the precise asymptotics in Davis law of large number for LPQD sequences. In literature [15], the auther discussed the precise asymptotics in Davis law of large number for PA sequences. LPQD weaker than PA sequences in dependencies, we discuss the precise asymptotics in Davis law of large numbers in weak dependence, and furtherly improve the existed results. |
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