A note on complete moment convergence for coordinatewise negatively associated random vectors in Hilbert spaces

Abstract

Let $\{X_n,n\ge 1\}$ be coordinatewise negatively associated random vectors taking values in real separable Hilbert space. In this note we prove the convergence of ${\sum }_{n=1}^{\infty }{n}^{-1-\alpha }E{({\max }_{1\le k\le n}|\left|S_k\right||-ϵn^{\alpha })}^{+}$ and ${\sum }_{n=1}^{\infty }{n}^{-1-\alpha }\hspace{0.17em}\text{log}\hspace{0.17em}nE{({\max }_{1\le k\le n}|\left|S_k\right||-ϵn^{\alpha })}^{+}$, where $S_k=X_1+\cdots +X_k.$

The present investigation provides the complete moment convergence for the case $\alpha r=1$ and generalizes some results in Huan (2015).

Keywords:

• Complete moment convergence
• coordinatewise negatively associated random vectors
• Hilbert spaces

2010 AMS SUBJECT CLASSIFICATION:

• 60F15
• 60B12

Acknowledgments

The author would like to sincerely thank the anonymous referees for their careful reading of the manuscript and valuable suggestions. This article was supported by Wonkwang University in 2019.