International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 4, Pages 745-752
On the almost sure convergence of weighted sums of random elements in
Department of Mathematics and Statistics, University of South Carolina, Columbia 29208, S. C., USA
Received 26 June 1980; Revised 17 February 1981
Copyright © 1981 R. L. Taylor and C. A. Calhoun. 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.
Let be a sequence of positive constants and where and . Let be a sequence of independent random elements in . The almost sure convergence of is established under certain integral conditions and growth conditions on the weights . 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).