Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables - Dug Hun Hong; Seok Yoon Hwang (International Journal of Mathematics and Mathematical Sciences)
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.