One-sided iterated logarithm laws of the form $\lim \sup (1/b_n) \sum^n_1 X_i = 1$, a.s. and $\lim \sup (1/b_n) \sum^n_1 X_i = -1$, a.s. are obtained for asymmetric independent and identically distributed random variables, the first when these have a vanishing but barely finite mean, the second when $E|X|$ is barely infinite. In both cases, $\lim \inf (1/b_n) \sum^n_1 X_i = -\infty$, a.s. The constants $b_n/n$ are slowly varying, decreasing to zero in the first case and increasing to infinity in the second. Although defined via the distribution of $|X|, b_n$ represents the order of magnitude of $E|\sum^n_1 X_i|$ when this is finite. Corresponding weak laws of large numbers are established and related to Feller's notion of "unfavorable fair games" and in the process a theorem playing the same role for the weak law as Feller's generalization of the strong law is proved.
"Iterated Logarithm Laws for Asymmetric Random Variables Barely with or Without Finite Mean." Ann. Probab. 5 (6) 861 - 874, December, 1977. https://doi.org/10.1214/aop/1176995656