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

One-sided iterated logarithm laws of the form lim (1/bₙ) ∑ⁿ₁ Xᵢ = 1, a.s. and lim (1/bₙ) ∑ⁿ₁ Xᵢ = -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/bₙ) ∑ⁿ₁ Xᵢ = -∞, a.s. The constants bₙ/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|, bₙ represents the order of magnitude of E|∑ⁿ₁ Xᵢ| 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.