A strong law for weighted sums of i.i.d. random variables

A strong law is proved for weighted sumsS _n=a _in X _i whereX _i are i.i.d. and {a _in} is an array of constants. When sup(n ^–1|a _in | ^q )^1/q <, 1<q andX _i are mean zero, we showE|X| ^p <,p ^l+q ^–1=1 impliesS _n /n 0. Whenq= this reduces to a result of Choi and Sung who showed that when the {a _in} are uniformly bounded,EX=0 andE|X|< impliesS _n /n 0. The result is also true whenq=1 under the additional assumption that lim sup |a _in |n ^–1 logn=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/< impliesS _n /n 0 for >1, but this is not true in general for 1/2<<1, even when theX _i are symmetric. In that case the additional assumption that (x ^1/ log^1/–1 x)P(|X|x)0 asx provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a _in}.