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, X; [set membership, variant] d} be a field of independent identically distributed real random variables, 0 < p < 2, and {a,; (, ) [set membership, variant] d - d, <= } a triangular array of real numbers, where d is the d-dimensional lattice. Under the minimal condition that sup, a, < [infinity], we show that - 1/p [summation operator] <= a, X --> 0 a.s. as --> [infinity] if and only if E(Xp(LX)d - 1) < [infinity] 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 [summation operator] <= a, X under the conditions that EX = 0, EX2 < [infinity], and E(X2(LX)d - 1/L2X) < [infinity] for almost all bounded families {a,; (, ) [set membership, variant] d - d, <= of numbers.