Complete convergence for randomly weighted sums of dependent random variables and an application

In this article, we establish complete convergence for randomly weighted sums of negatively orthant-dependent random variables. We also establish complete convergence for the maximums of randomly weighted sums of negatively associated random variables. As a corollary, a Marcinkiewicz-Zygmund-type strong law for randomly weighted sums of negatively associated random variables is obtained. The results can be applied to the bootstrap sample means, and an open problem in a convergence rate of the bootstrap sample means posed by Csorgo (2004) is solved completely.