The Study Of Convergence For Weighted Sums Of NOD Random Variables

The limit theory of dependent sequences is one of the central issues for studying prob-ability, which has very wide application in fields such as multivariate statistical analysis, economic decisions and insurance actuarial science, reliability, weather forecast, survival analysis, engineering and technology and so on. In this article, we investigate conver-gence properties for weighted sums of negatively orthant dependent random variables by several techniques such as Borel-Cantelli lemma, Kronecker lemma, Markov’s inequali-ty, Holder’s inequality, Jensen’s inequality, Cr inequality, Rosenthal inequality, maximal inequality, the truncated method for random variables, etc., and obtain several new result-s, for example, the Marcinkiewicz-Zygmund-type strong law of large numbers, complete convergence, complete moment convergence, and so on. our results improve and generalize the corresponding results of the cited references.Firstly, based on the moment inequality and maximal inequality for weighted sums of NOD random variables, we obtain Marcinkiewicz-Zygmund-type strong law of large numbers for weighted sums of NOD random variables. As a result, we improve the corre-sponding results in Bai and Cheng[7], namely from independent and identically distributed random variables to NOD random variables without adding any other conditions. Besides, by the truncated method, we also obtain strong convergence for Jamison-type weighted sums of NOD random variables, which generalize the corresponding results in Wang[51].Secondly, with techniques of maximal inequality and truncated method, we pay much attention to complete convergence of weighted sums for arrays of rowwise NOD random variables. Our results not only generalize the case of independent and identically distribut-ed random variables in Baum and Katz[6], but also establish Marcinkiewicz-Zygmund-type strong law of large numbers for weighted sums of NOD random variables.Finally, we study the complete moment convergence for double-indexed weighted sums of arrays of rowwise NOD random variables. It generalizes the case for weighted sums of NOD random variables, and it reveals the relationship of complete convergence in the complete moment converging process. Meanwhile, Marcinkiewicz-Zygmund-type strong law of large numbers for weighted sums of NOD random variables has also been established, which generalizes the corresponding results of independent and identically distributed random variables and NA random variables.