International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1983 / Article

Open Access

Volume 6 |Article ID 657129 | https://doi.org/10.1155/S0161171283000046

Robert Lee Taylor, "Complete convergence for weighted sums of arrays of random elements", International Journal of Mathematics and Mathematical Sciences, vol. 6, Article ID 657129, 11 pages, 1983. https://doi.org/10.1155/S0161171283000046

Complete convergence for weighted sums of arrays of random elements

Received22 Jul 1982

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=1exp(α/An)< for each α ϵ R+ where An=k=1ank2. The complete convergence of k=1ankXnk 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.

Copyright © 1983 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.


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