International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 4, Pages 805-814
doi:10.1155/S0161171287000899

Strong laws of large numbers for arrays of rowwise independent random elements

Robert Lee Taylor1 and Tien-Chung Hu2

1Depatment of statistics , University of Georgia, Athens 30602, GA, USA
2Depatment of Mathematics, National Tsing-Hua University, Hsin-chu, Taiwan

Received 18 November 1986

Copyright © 1987 Robert Lee Taylor and Tien-Chung Hu. 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} be an array of rowwise independent random elements in a separable Banach space of type p+δ with EXnk=0 for all k, n. The complete convergence (and hence almost sure convergence) of n−1/p∑k=1nXnk to 0, 1≤p<2, is obtained when {Xnk} are uniformly bounded by a random variable X with E|X|2p<∞. When the array {Xnk} consists of i.i.d, random elements, then it is shown that n−1/p∑k=1nXnk converges completely to 0 if and only if E‖X11‖2p<∞.