International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 1, Pages 171-177
doi:10.1155/S0161171299221710
Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables
1School of Mechanical and Automotive Engineering, Catholic University of Taegu-Hyosung, Kyungbuk 712-702, South Korea
2Department of Mathematics, Taegu University, Kyungbuk 713-714, South Korea
Received 31 May 1996
Copyright © 1999 Dug Hun Hong and Seok Yoon Hwang. 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 {Xij} be a double sequence of pairwise independent random variables. If P{|Xmn|≥t}≤P{|X|≥t} for all nonnegative real numbers t and E|X|p(log+|X|)3<∞, for 1<p<2, then we prove that ∑i=1m∑j=1n(Xij−EXij)(mn)1/p→0 a.s. as m∨n→∞. (0.1) Under the weak condition of E|X|plog+|X|<∞, it converges to 0 in L1. And the results can be generalized to an r-dimensional array of random variables under the conditions E|X|p(log+|X|)r+1<∞,E|X|p(log+|X|)r−1<∞, respectively, thus, extending Choi and Sung's result [1] of the one-dimensional case.