Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of random variables with infinite moment and its applications

Abstract

In this work, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of widely orthant dependent random variables is established under mild conditions including infinite variance. As applications, some new results such as the Cesàro strong law of large numbers, the strong consistency of the least squares estimator in multiple linear regression models, and the strong consistency of the wavelet estimator in nonparametric regression models are presented. Simulation studies are also provided to support the theoretical results.

Keywords:

• Strong law of large numbers
• widely orthant dependent
• infinite moment
• consistency
• least squares estimator
• multiple linear regression model
• wavelet estimator
• nonparametric regression model

Mathematical Subject Classifications:

• 60F15
• 62G05
• 62G20

Acknowledgements

The authors are most grateful to the Editor and anonymous referees 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 author(s).

Additional information

Funding

Supported by the National Social Science Foundation of China (22BTJ059), the National Natural Science Foundation of China (11871072, 12201079), the Natural Science Foundation of Anhui Province (2108085MA06, 2108085QA15), and the National Fund Cultivation Project of Chizhou University (CZ2021GP04).