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International Journal of Mathematics and Mathematical Sciences
Volume 4, Issue 4, Pages 745-752
http://dx.doi.org/10.1155/S0161171281000574

On the almost sure convergence of weighted sums of random elements in D[0,1]

Department of Mathematics and Statistics, University of South Carolina, Columbia, S. C. 29208, USA

Received 26 June 1980; Revised 17 February 1981

Copyright © 1981 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 {wn} be a sequence of positive constants and Wn=w1++wn where Wn and wn/Wn. Let {Wn} be a sequence of independent random elements in D[0,1]. The almost sure convergence of Wn1k=1nwkXk is established under certain integral conditions and growth conditions on the weights {wn}. The results are shown to be substantially stronger than the weighted sums convergence results of Taylor and Daffer (1980) and the strong laws of large numbers of Ranga Rao (1963) and Daffer and Taylor (1979).