Convergence properties for weighted sums of weakly dependent random vectors in Hilbert spaces

Abstract

In this paper, some results on the complete moment convergence, complete convergence, complete integral convergence, and strong law of large numbers for weighted sums of weakly dependent random vectors in Hilbert spaces are established. The results obtained in this paper improve and extend the corresponding ones of Sung [Complete convergence for weighted sums of ${\rho }^{\ast }$-mixing random variables, Discrete Dyn. Nat. Soc. 2010 (2010), Article ID 630608, 13 pages], Huan [Baum-Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces, Acta Math. Hungar. 144(1) (2014), pp. 132–149], Huan [On complete convergence for sequences of random vectors in Hilbert spaces, Acta Math. Hungar. 147(1) (2015), pp. 205–219], and Ko [The complete moment convergence for CNA random vectors in Hilbert spaces, J. Inequal. Appl. 2017 (2017), Article ID 290, 11 pages].

Keywords:

• Weakly dependent random vector
• complete moment convergence
• complete convergence
• complete integral convergence
• strong law of large numbers

Acknowledgements

The authors are most grateful to the Editor and anonymous referee for carefully reading the manuscript and 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

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