A strong law for weighted sums of IID random variables

A strong law is proved for weighted sums S-n = Sigma a(in)X(i) where X(i) are i.i.d. and {a(in)} is an array of constants. When sup(n(-1) Sigma\a(in)\(q))(1/q) < infinity, 1 < q less than or equal to infinity and X(i) are mean zero, we show E\X\(p) < infinity, p(-1) + q(-1) = 1 implies S-n/n -->(a.s) 0. When q = infinity this reduces to a result of Choi and Sung((1)) who showed that when the {a(in)} are uniformly bounded, EX = 0 and E\X\ < infinity implies S-n/n -->a.s. 0. The result is also true when q = 1 under the additional assumption that lim sup \a(i)n\ n(-1) log n = 0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a(in)} are uniformly bounded, E\X\(1/alpha) < infinity implies S-n/n(alpha) -->(a.s.) 0 for alpha > 1, but this is not true in general for 1/2 < alpha < 1, even when the X(i) are symmetric. In that case the additional assumption that (x(1/alpha)log(1/alpha-1 x) P(\X\ greater than or equal to x) --> 0 as x up arrow infinity provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a(in)}.