The set of limit points associated with a generalized iterated logarithm law for sums of asymmetric i.i.d. random variables is shown to be the entire closed interval bounded by the $\lim \inf$ and the $\lim \sup$.
"Limit Points Associated with Generalized Iterated Logarithm Laws." Ann. Probab. 8 (2) 390 - 392, April, 1980. https://doi.org/10.1214/aop/1176994787