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=1mj=1n(XijEXij)(mn)1/p0a.s.asmn.(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|)r1<, respectively, thus, extending Choi and Sung's result [1] of the one-dimensional case.