Iterated Logarithm Laws for Asymmetric Random Variables Barely with or Without Finite Mean

Abstract

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.