A triumvirate of sufficient conditions is given for unbounded, independent random variables to obey the Law of the Iterated Logarithm (LIL). As special cases, new results for weighted i.i.d. random variables and the Hartman-Wintner theorem are obtained. Necessity of finite variance for the two-sided LIL is shown to carry over for a large class of weighted i.i.d. random variables and the Marcinkiewicz-Zygmund example is generalized, simplified and clarified.
"On the Law of the Iterated Logarithm." Ann. Probab. 2 (4) 714 - 728, August, 1974. https://doi.org/10.1214/aop/1176996614