International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 3, Pages 443-450
doi:10.1155/S0161171297000604
Convergence in mean of weighted sums of {ank}-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 M. Ordóñez Cabrera. 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.
Abstract
The convergence in mean of a weighted sum ∑kank(Xk−EXk) 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.