On the law of large numbers and arithmetic functions

In this paper we give an optimal condition for the strong law of large numbers limN→∞∑n=1Nf(n)Xn∑n=1Nf(n)=a.s.EX where X,X1,X2,…X,X1,X2,… are i.i.d. integrable random variables and ff is an additive arithmetic function. We also give sufficient criteria for multiplicative weight functions ff.