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