Weak and one-sided strong laws for random variables with infinite mean

Let a_j be positive weight constants and X_j be independent non-negative random variables (j=1,2,…) and S_n(a)=∑_i=1 ⁿa_iX_i. If the X_j have the same relatively stable distribution, then under mild conditions there exist constants b_n→∞ such that W¯_n(a)=b_n ⁻¹S_n(a)→p1, i.e., a weak law of large numbers holds. If the weights comprise a regularly varying sequence, then under some additional technical conditions, this outcome can be strengthened to a strong law if and only if the index of regular variation is −1. This paper addresses a case where the X_j are not identically distributed, but rather the tail probability P(a_jX_j>x) is asymptotically proportional to a_j(1−F(x)), where F is a relatively stable distribution function. Here the weak law holds but the strong law does not: under typical conditions almost surely lim inf_n→∞W¯_n(a)=1 and lim sup_n→∞W¯_n(a)=∞.