Some universal results on the behavior of increments of partial sums

Abstract

We establish very general one-sided results on the lim sup behavior of increments of suitably normalized partial sums of i.i.d. random variables. Our main results apply to arbitrary nondegenerate positive random variables which need not have any finite moments. As a corollary we can show that such results also hold for not necessarily positive random variables whose negative parts have finite moment-generating functions.