On the Strong Law of Large Numbers and the Law of the Logarithm for Weighted Sums of Independent Random Variables with Multidimensional Indices

Let (X, Xn; n ∈ Nd} be a field of independent identically distributed real random variables, 0 < p < 2, and {an,k; (n, k) ∈ Nd × Nd, k ≤ n} a triangular array of real numbers, where Nd is the d-dimensional lattice. Under the minimal condition that supn, k |an, k| < ∞, we show that |n|− 1/p ∑k ≤ nan, kXk → 0 a.s. as |n| → ∞ if and only if E(|X|p(L|X|)d − 1) < ∞ provided d ≥ 2. In the above, if 1 ≤ p < 2, the random variables are needed to be centered at the mean. By establishing a certain law of the logarithm, we show that the Law of the Iterated Logarithm fails for the weighted sums ∑k ≤ nan, kXk under the conditions that EX = 0, EX2 < ∞, and E(X2(L|X|)d − 1/L2|X|) < ∞ for almost all bounded families {an,k; (n, k) ∈ Nd × Nd, k ≤ n of numbers.