A law of the iterated logarithm is proved using an extension of the Borel-Cantelli lemma and the asymptotic expansion of the left tail of this stable law.
J. L. Mijnheer. "A Law of the Iterated Logarithm for the Asymmetric Stable Law with Characteristic Exponent One." Ann. Math. Statist. 43 (1) 358 - 362, February, 1972. https://doi.org/10.1214/aoms/1177692732