Expansion of transition distributions of Lévy processes in small time

Abstract

We establish for small time t a series expansion of the transition density and the transition distribution function of Lévy processes in terms of the density and the spectral function of the Lévy measure, respectively. Furthermore, the integrals ith respect tothe distribution function weighted by 1/t are proved to converge to the integral ith respect tothe spectral function of the Lévy measure, when the integrated function does not increase too fast and t → 0.