Using suitable self-normalizations for partial sums of i.i.d. random variables, a law of the iterated logarithm, which generalizes the classical LIL, is proved for all distributions in the Feller class. A special case of these results applies to any distribution in the domain of attraction of some stable law.
Philip S. Griffin. James D. Kuelbs. "Self-Normalized Laws of the Iterated Logarithm." Ann. Probab. 17 (4) 1571 - 1601, October, 1989. https://doi.org/10.1214/aop/1176991175