International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 1, Pages 69-79
doi:10.1155/S0161171283000046
Complete convergence for weighted sums of arrays of random elements
Department of Mathematics and Statistics, University of South Carolina, Columbia 29208, S. C., USA
Received 22 July 1982
Copyright © 1983 Robert Lee Taylor. 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 {Xnk:k,n=1,2,…} be an array of row-wise independent random elements in a separable Banach space. Let {ank:k,n=1,2,…} be an array of real numbers such that ∑k=1∞|ank|≤1 and ∑n=1∞exp(−α/An)<∞ for each α ϵ R+ where An=∑k=1∞ank2. The complete convergence of ∑k=1∞ankXnk is obtained under varying moment and distribution conditions on the random elements. In particular, laws of large numbers follow for triangular arrays of random elements, and consistency of the kernel density estimates is obtained from these results.