A note on the rate of complete convergence for weighted sums of arrays of Banach space valued random elements. (English) Zbl 1217.60007

Sung, Soo Hak; Volodin, Andrei

Stochastic Anal. Appl. 29, No. 2, 282-291 (2011).

Let \((X_{ni})\) be an array of rowwise independent random variables in a real separable Banach space and \((a_{ni})\) be real numbers. S. E. Ahmed, R. Giuliano Antonini and A. Volodin [Stat. Probab. Lett. 58, No. 2, 185–194 (2002; Zbl 1017.60013)] proved that under some special conditions (including conditions on a scalar \(\beta\)) for all \(\varepsilon>0\)
\[ \sum_{n=1}^\infty n^\beta P \left(\left\|\sum_{i=1}^\infty a_{ni }X_{ni}\right\|> \varepsilon\right)<\infty \]
(complete convergence). In the paper under review, the authors improve and complement this result, using a simpler method.

Reviewer: Anatolij M. Plichko (Krakow)

MSC:

60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05	Central limit and other weak theorems
60F15	Strong limit theorems

Keywords:

array of random elements; complete convergence; convergence in probability; rowwise independence; weighted sums

Citations:

Zbl 1017.60013

References:

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