Complete Convergence And L~r Convergence For Extended Negatively Dependent Random Variables

The limit theory of dependent sequences is one of the central issues for studying probability, which is widely applied in fields such as multivariate sta-tistical analysis, economic decisions, actuary and reliability, weather forecast, survival analysis, engineering and so on. In this article, we investigate com-plete and Lτ convergence properties of extended negatively dependent random variables by several techniques such as stochastic domination, index inequality, Markov’s inequality, CT inequality, Rosenthal inequality, the truncated method for random variables, etc., obtaining several new results. Our main results im-prove and generalize the corresponding results of the cited references.Firstly, we study the complete convergence for weighted sums of extend-ed negatively dependent random variable sequences. On the basis of proper weight coefficient and moment condition, we obtain complete convergence for weighted sums of extended negatively dependent by index inequality, stochas-tic domination, and the truncated method for random variables, which greatly generalize the corresponding results in Zarei and Jabbari.Secondly, the complete convergence of arrays of rowwise extended neg-atively dependent random variables is obtained by using the Rosenthal type moment inequality. Some sufficient conditions to prove the complete conver-gence are provided. In addition, a necessary condition to prove the complete convergence is presented. The results extend identically distributed negatively dependent random variables in Wu to nonidentically distributed extended negatively dependent random variables.Finally, we investigate the Lτ convergence for weighted sums of extended negatively dependent random variables under r-th uniform integrability by Rosenthal inequality, which generalize the corresponding results in Chen.