Higher variations for free L\'evy processes

Abstract

For a general free L\'evy process, we prove the existence of its higher variation processes as limits in distribution, and identify the limits in terms of the L\'evy-It\^o representation of the original process. For a general free compound Poisson process, this convergence holds in probability. This implies joint convergence in distribution to a $k$-tuple of higher variation processes, and so the existence of $k$-fold stochastic integrals as limits in probability. If the existence of moments of all orders is assumed, the result holds for free additive (not necessarily stationary) processes and more general approximants. In the appendix we note relevant properties of symmetric polynomials in non-commuting variables.