THE LAW OF LARGE NUMBERS FOR PRODUCT PARTIAL SUM PROCESSES INDEXED BYSETS

Let N = {1, 2, ...) and let {X(i):i is-an-element-of N(d1)} and {Y(j): j is-an-element-of N(d2)} be two families of i.i.d. integrable random variables. Let S(nA) be the sum of those X(i)Y(j)'s for which A subset -of [0,1]d, d = d1 + d2 and (i/n,j/n) is-an-element-of A. It is proved that S(.) satisfies a strong law of large numbers that is uniform ove r A, where A is a family of subsets of [0, 1]d satisfying some conditi ons. (C) 1994 Academic Press, Inc.