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 row-wise 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
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<∞.