Let (\left{X, X_{n} ; \vec{n} \in \mathbb{N}^{d}\right}) be a field of independent identically distributed real random variables, (0<p<2), and (\left{a_{\bar{n}, \bar{k}} ;(\bar{n}, \bar{k}) \in \mathbb{N}^{d} \times \mathbb{N}^{d}, \bar{k} \leqslant \bar{n}\right}) a triangular array of real numbers, where (\mathbb{N}^{d}) is the (d)-dimensional lattice. Under the minimal condition that (\sup {n, k}\left|a{n, \bar{k}}\right|<\infty), we show that (|\bar{n}|^{-1 / p} \sum_{\bar{k} \leqslant \bar{n}} a_{\bar{n}, \bar{k}} X_{\bar{k}} \rightarrow 0) a.s. as (|\bar{n}| \rightarrow \infty) if and only if (E\left(|X|^{p}(L|X|)^{d}{ }^{1}\right)<\infty) provided (d \geqslant 2). In the above, if (1 \leqslant 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 (\sum_{k \leqslant i n} a_{\bar{n}, k} X_{k}) under the conditions that (E X=0, E X^{2}<\infty), and (E\left(X^{2}(L|X|)^{d-1} / L_{2}|X|\right)<\infty) for almost all bounded families (\left{a_{\bar{n}, \bar{k}} ;(\bar{n}, \bar{k}) \in \mathbb{N}^{d} \times \mathbb{N}^{d}\right.), (\bar{k} \leqslant \bar{n}}) of numbers. 1995 Academic Press, Inc.