Research On Inequalities And Strong Convergence Properties For Some Dependence Sequences

Probability Limit Theory is one of the important branches of Probability and Statistics. A research direction of modern Probability Limit Theory is to weaken the restrictions of independence, so as to be closer to reality and to make for application. Due to the practical problems demand, people pay more and more attention on dependent random variable sequences. Limit theory of dependent sequences have very wide application in many fields such as survival analysis, insurance and financial mathematics, complex system, multivariate statistical analysis, reliability theory, making economic decisions and so on.In this paper, we mainly study some probability moment inequalities for dependent sequences. The strong limit theorems for these sequences are stud-ied based on these inequalities and lemmas. Such as Markov inequalities, Cr inequalities, Rosenthal inequalities, Borel-Cantelli lemma, Kronecker lemma, and so on. In chapter one, we briefly introduce the background of thesis re-search, the definitions of several dependent random variable sequences and the relationships among them. The definition of the maximal moment inequali-ty with exponent 2 and some lemmas are also needed. In chapter two, since END random variables are much weaker than independent random variable sequences, NA sequences and NOD sequences, it is interesting for us to s-tudy the limit behavior of END sequences. By the Rosenthal type-moment inequalities of END sequences, we further study the Khintchine-Kolmogorov type convergence theorem. Some strong limit results for weighted sums are obtained, which generalize the corresponding results for independent random variable sequences, negatively associated random variable sequences and NOD random variable sequences. In chapter three, we study the strong law of large numbers(SLLN) for a class of random variables, satisfying the maximal mo-ment inequality with exponent 2. Our results embrace the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for this class of random variables. In addition, strong growth rate for weighted sums of this class of random variables is presented. Moreover, our results extend the corresponding results in the literature.