Vol. 21, No. 1, 1967

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Sample function behavior of increasing processes with stationary, independent increments

Bert E. Fristedt

Vol. 21 (1967), No. 1, 21–33

In this paper we are concerned with the sample functions of increasing stochastic processes, Xν, having stationary, independent increments; normalized so that Xν has no deterministic linear component and Xν(0) = 0, (i.e., X is a subordinator). For h a fixed function, we are interested in the following two events: { ω : Xν(t,ω) > h(t) infinitely often as t 0}, { ω : Xν(t,ω) > h(t) infinitely often as t →∞}. In case Xν is a stable process, Khinchin has given integral tests to apply to a wide class of hs in order to decide whether one, the other, or both of these two events have probability zero or one. The purpose of this paper is to give similar results, without assuming Xu to be stable.

Mathematical Subject Classification
Primary: 60.60
Received: 18 November 1965
Published: 1 April 1967
Bert E. Fristedt