We obtain a generalization of part "if" of the Erdos-Wintner theorem to additive arithmetic functions with values in certain Banach spaces. Some applications in asymptotic density are considered.
Jesus de la Cal. "On the Three Series Theorem in Number Theory." Ann. Probab. 17 (1) 357 - 361, January, 1989. https://doi.org/10.1214/aop/1176991514