The Ito-Nisio Theorem is a powerful tool that enables to deduce the uniform convergence from the pointwise convergence in Karkhunen-Loeve-type series expansions. Unfortunately, this tool fails for stronger than uniform modes of convergence, such as Lipschitz or p-variation convergence. On the other hand, the latter mode is natural in the study of processes with jumps. In this work we give a framework for and establish a generalization of the Ito-Nisio Theorem that covers also strong modes of convergence.

For applications to Levy processes, we find the optimal subspace of functions of finite phi-variation where all Levy processes without Gaussian component live and admit strongly converging series expansions. This yields an extension of the celebrated S.J. Taylor's theorem to Levy processes. We apply these results to Levy driven SDEs and infinitely divisible random fields.

This talk is based on a joint work with A. Basse-O'Connor and J. Hoffmann-Jorgensen.