If $X$ takes values in a Banach space $B$ and is in the domain of normal attraction of a Gaussian law on $B$ with $EX = 0, E(\|X\|^2/L_2\|X\|) < \infty$, then it is known that $X$ satisfies the compact law of the iterated logarithm as described in Goodman, Kuelbs and Zinn , Theorem 4.1. In this paper the analogous result is demonstrated when $X$ is merely in the domain of attraction of a Gaussian law. The functional LIL is also obtained in this setting. These results refine Corollary 7 of Kuelbs and Zinn , as well as various functional LILs.
J. Kuelbs. "The LIL when $X$ is in the Domain of Attraction of a Gaussian Law." Ann. Probab. 13 (3) 825 - 859, August, 1985. https://doi.org/10.1214/aop/1176992910