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International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 3, Pages 443-450

Convergence in mean of weighted sums of {an,k}-compactly uniformly integrable random elements in Banach spaces

Department of Mathematical Analysis, University of Sevilla, Sevilla, Spain

Received 23 October 1995; Revised 14 March 1996

Copyright © 1997 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The convergence in mean of a weighted sum kank(XkEXk) of random elements in a separable Banach space is studied under a new hypothesis which relates the random elements with their respective weights in the sum: the {ank}-compactly uniform integrability of {Xn}. This condition, which is implied by the tightness of {Xn} and the {ank}-uniform integrability of {Xn}, is weaker than the compactly miform integrability of {Xn} and leads to a result of convergence in mean which is strictly stronger than a recent result of Wang, Rao and Deli.