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International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 11, Pages 789-794
http://dx.doi.org/10.1155/S0161171200003112

Complete convergence for sums of arrays of random elements

Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea

Received 13 October 1997

Copyright © 2000 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 {Xni} be an array of rowwise independent B-valued random elements and {an} constants such that 0<an. Under some moment conditions for the array, it is shown that i=1nXni/an converges to 0 completely if and only if i=1nXni/an converges to 0 in probability.