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

Abstract

Let aj be positive weight constants and Xj be independent non-negative random variables (j=1,2,…) and Sn(a)=∑i=1naiXi. If the Xj have the same relatively stable distribution, then under mild conditions there exist constants bn→∞ such that W¯n(a)=bn−1Sn(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 Xj are not identically distributed, but rather the tail probability P(ajXj>x) is asymptotically proportional to aj(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 infn→∞W¯n(a)=1 and lim supn→∞W¯n(a)=∞.