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

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.