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Acta Mathematica Hungarica

, Volume 147, Issue 1, pp 205–219 | Cite as

On the complete convergence for sequences of random vectors in Hilbert spaces

  • N. V. Huan
Article

Abstract

This note is a continuation of the paper [7]. Let \({\{X_{n},\ {n\geqq 1}\}}\) be a sequence of coordinatewise negatively associated random vectors taking values in a real separable Hilbert space with the k-th partial sum S k , \({k \geqq 1}\). We provide conditions for the convergence of \({\sum_{n=1}^{\infty} \frac{1}{n} \, \mathbb{P}({\rm max}_{1 \leqq k \leqq n} \|S_k\| > \varepsilon n^{\alpha })}\) and \({\sum_{n=1}^{\infty} \frac{{\rm log} n}{n} \, \mathbb{P}({\rm max}_{1 \leqq k \leqq n} \|S_k\| > \varepsilon n^{\alpha})}\) for all \({\varepsilon > 0}\). The converses of these results are also discussed.

Key words and phrases

complete convergence Baum–Katz theorem coordinatewise negatively associated random vector Hilbert space 

Mathematics Subject Classification

60F15 60B12 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  1. 1.Department of Mathematics and ApplicationsSaigon UniversityHo Chi Minh CityVietnam

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