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

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.