Sample function behavior of increasingprocesses with stationary, independent increments

In this paper we are concerned with the sample functionsof increasing stochastic processes, Xv, having stationary, independent increments; normalized so that Xv has no deterministiclinear component and X_v(0) = 0, (i.e., Xv is a subordinator).two events:{ω: Xv(t, ω) > h(t) infinitely often as t →0}, {ω: Xu(tf ω) > h(t) infinitely often as t → ∞} .In case Xv is a stable process, Khinchin has given integraltests to apply to a wide class ofh's in order to decide whetherone, the other, or both of these two events have probability zeroor one. The purpose of this paper is to give similar results, without assuming X to be stable.