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

A strong law is proved for weighted sumsS_n=a_inX_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/n0. 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/n0. 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/n0 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/–1x)P(|X|x)0 asx provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a_in}.