Almost sure limit theorems for stable distributions

Consider a sequence of i.i.d. random variables in the domain of attraction of a stable distribution with an exponent in (0,2]. A universal result in almost sure limit theorem for the partial sums is established. Our results substantially extend and improve those on the almost sure central limit theorem previously obtained by Jonsson 2007, Berkes and Csáki 2001, and Hörmann 2007.