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International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 3, Pages 559-568
http://dx.doi.org/10.1155/S0161171299225598

Stability for randomly weighted sums of random elements

1Department of Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan
2Institute of Mathematics, Academic Sinica, Taipei 11529, Taiwan

Received 12 May 1998

Copyright © 1999 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.

Abstract

Let {Xn:n=1,2,3,} be a sequence of i.i.d. random elements taking values in a separable Banach space of type p and let {An,i:i=1,2,3,;n=1,2,3,} be an array of random variables. In this paper, under various assumptions of {An,i}, the necessary and sufficient conditions for i=1An,iXi0 a.s. are obtained. Also, the necessity of the assumptions of {An,i} is discussed.