Complete q-th moment convergence and its application in the dependent bootstrap

Abstract

Let be an array of rowwise pairwise negatively quadrant dependent random variables with mean zero and as , where is a sequence of positive integers. The complete qth moment convergence results for and are established which are much stronger than the complete convergence results obtained in the literature. As applications of our main results, we obtain some complete moment theorems for the dependent bootstrap samples arising from a sequence of independent and identically distributed random variables.

Keywords:

• Pairwise negatively quadrant dependent random variables
• complete moment convergence
• dependent bootstrap
• sums of identically distributed random variables

Acknowledgements

The authors are most grateful to the Editor-in-Chief, Associate Editor and anonymous referee for carefully reading the manuscript and for valuable suggestions which helped in improving an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11671012], [grant number 11501004], [grant number 11501005]; and the Natural Science Foundation of Anhui Province [grant number 1508085J06].