On the oscillation of infinitely divisible and some other processes

A sufficient condition is given for processes admitting a series expansion with partially dependent components to have nonrandom oscillation. Included are infinitely divisible processes with no Gaussian component. This immediately gives information about the sample paths of such processes, e.g. a Belayev type dichotomy between path continuity and unboundedness for stationary or self-similar processes. The sufficient condition for nonrandom oscillation is shown to be necessary for a large class of infinitely divisible processes to have finite nonrandom oscillation. It is also used to relate path continuity to continuity at each point. Similar results are described for path differentiability.