Complete convergence for maximum of weighted sums of WNOD random variables and its application

Abstract

In this paper, the complete convergence for maximum of weighted sums of widely negative orthant dependent (WNOD) random variables are investigated. Some sufficient conditions for the convergence are provided and a relationship between the weight and the boundary function is revealed. Additionally, a Marcinkiewicz-Zygmund type strong law of large number for weighted sums of WNOD random variables is obtained. The results obtained in this paper generalize some corresponding ones for independent and some dependent random variables. As an application, the strong consistency for the weighted estimator in a non-parametric regression model is established. MR(2010) Subject Classification: 60F15; 62G05.

Keywords:

• Complete convergence
• widely negative orthant dependent
• maximum of weighted sums
• strong law of large numbers

Acknowledgements

The authors are most grateful to the editor and anonymous referees for their careful reading of the manuscript and many valuable suggestions which helped to improve an earlier version of this paper.

Additional information

Funding

This research was partially supported by the National Natural Science Foundation of China (No. 11571250, 71803140).