Equivalent conditions of complete moment convergence for randomly weighted sums of random variables and some applications with random design

In this paper, the equivalent conditions of complete moment convergence for maximal randomly weighted sums of negatively associated random variables with negatively associated random weights are investigated through using the slicing and monotonic truncation methods. Furthermore, some corollaries about the complete convergence and the almost sure convergence for random variables with randomly weighted are presented. The results obtained in this paper remain valid for some dependent random variables and mixing random variables, serving as generalizations and improvements of some known ones. Additionally, the derived results are applied to the bootstrap sample means, yielding the complete moment convergence and complete convergence for the bootstrap sample means of negatively associated random variables. These results are further applied to a simple linear errors-in-variables regression model with random design, establishing the strong consistency of the least squares estimator. Finally, a numerical simulation is conducted to evaluate the performances in finite samples.