Abstract: | Let (X, X
;
d} be a field of independent identically distributed real random variables, 0 < p < 2, and {a
,
; (
,
)
d ×
d,
≤
} a triangular array of real numbers, where
d is the d-dimensional lattice. Under the minimal condition that sup
,
|a
,
| < ∞, we show that |
|− 1/p ∑
≤
a
,
X
→ 0 a.s. as |
| → ∞ 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 ∑
≤
a
,
X
under the conditions that EX = 0, EX2 < ∞, and E(X2(L|X|)d − 1/L2|X|) < ∞ for almost all bounded families {a
,
; (
,
)
d ×
d,
≤
of numbers. |