On the uniform convergence of random series in Skorohod space and representations of cadlag infinitely divisible processes

The Ito-Nisio theorem implies that various series expansions of a Brownian motion, and of other sample continuous Gaussian processes, converge uniformly pathwise, which was the original motivation for the theorem. To facilitate the study of sample discontinuous infinitely divisible processes, we extend the Ito-Nisio Theorem to the Skorohod space of cadlag functions equipped with the uniform norm. It is known that the Ito-Nisio theorem fails in many non-separable Banach spaces, which shows that, in some sense, the Skorohod space under the uniform norm is exceptional.  

We illustrate our results on an example of Volterra-type stable processes. We obtain explicit representations of the process of jumps, and of related path functionals, in a general non-Markovian setting. To this aim we obtain new criteria for such processes to have cadlag modifications, which may also be of independent interest.

This talk is based on a joint work with Andreas Basse-O'Connor.