×

zbMATH — the first resource for mathematics

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

MSC:
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1016/S0167-7152(02)00126-8 · Zbl 1017.60013 · doi:10.1016/S0167-7152(02)00126-8
[2] DOI: 10.4134/JKMS.2006.43.4.815 · Zbl 1112.60003 · doi:10.4134/JKMS.2006.43.4.815
[3] Chen P., Siberian Adv. Math. 16 pp 1– (2006)
[4] DOI: 10.1007/s10959-007-0118-6 · Zbl 1139.60007 · doi:10.1007/s10959-007-0118-6
[5] DOI: 10.1073/pnas.33.2.25 · Zbl 0030.20101 · doi:10.1073/pnas.33.2.25
[6] Hu T.-C., Teor. Veroyatnost. i Primenen. 47:533–547. [translation in Theory Probab. Appl. 47 pp 455– (2002)
[7] DOI: 10.1155/S0161171294000013 · Zbl 0798.60006 · doi:10.1155/S0161171294000013
[8] DOI: 10.1080/07362999708809474 · Zbl 0902.60011 · doi:10.1080/07362999708809474
[9] Volodin A., Lobachevskii J. Math. 15 pp 21– (2004)
[10] DOI: 10.1080/07362999308809305 · Zbl 0764.60037 · doi:10.1080/07362999308809305
[11] DOI: 10.1080/07362999908809645 · Zbl 0940.60032 · doi:10.1080/07362999908809645
[12] DOI: 10.1214/aop/1176994517 · Zbl 0449.60002 · doi:10.1214/aop/1176994517
[13] DOI: 10.1214/aop/1176995149 · Zbl 0399.60007 · doi:10.1214/aop/1176995149
[14] DOI: 10.1016/j.spl.2009.03.001 · Zbl 1168.60337 · doi:10.1016/j.spl.2009.03.001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.