Complete and complete moment convergence for randomly weighted sums of ρ*-mixing random variables and its applications

ABSTRACT

In this paper, we will study the complete and complete moment convergence for double-indexed randomly weighted sums of ${\rho }^{\ast }$-mixing random variables. Several sufficient conditions to prove the complete and complete moment convergence for randomly weighted sums of ${\rho }^{\ast }$-mixing random variables are presented. The results obtained in this paper extend some corresponding ones in the literature. As applications, we further study the convergence of the state observers of linear-time-invariant systems and the complete consistency for the weighted estimator in nonparametric regression models based on ${\rho }^{\ast }$-mixing random errors. Finally, some numerical simulations are provided to verify the validity of theoretical results.

KEYWORDS:

• ${\rho }^{\ast }$-mixing random variables
• randomly weighted sums
• complete moment convergence
• complete convergence
• linear-time-invariant system
• nonparametric regression model

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 60F15
• 60G20
• 93E24

Acknowledgements

The authors are most grateful to the Editor and two anonymous referees for carefully reading the manuscript and valuable suggestions which helped in significantly improving an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

Supported by the National Natural Science Foundation of China (11671012, 11871072, 11701004, 11701005), the Natural Science Foundation of Anhui Province (1808085QA03, 1908085QA01, 1908085QA07), the Provincial Natural Science Research Project of Anhui Colleges (KJ2019A0003) and the Project on Reserve Candidates for Academic and Technical Leaders of Anhui Province (2017H123).