The Law of Large Numbers for Product Partial Sum Processes Indexed by Sets

Let N = {1, 2, ...} and let {Xi:i ∈ Nd1} and {Yj:j ∈ Nd2} be two families of i.i.d. integrable random variables. Let S(nA) be the sum of those XiYj′s for which A ⊂ [0,1]d, d = d1 + d2 and (i/n,,j/n) ∈ A. It is proved that S(·) satisfies a strong law of large numbers that is uniform over A, where A is a family of subsets of [0, 1]d satisfying some conditions.