Mixing of generating functionals and applications to (semi-)stability of probabilities on groups

Abstract

Let (X_t) be a Lévy process on a simply connected nilpotent Lie group with corresponding continuous convolution semigroup (v_t). Assume (v_t) to be semistable. Then a suitable mixing of (v_t) resp. a random time substitution of (X_t) belongs to the domain of attraction of a stable Lévy process (U_t), the infinitesimal generator resp. the generating functional of which is representable as mixing of semistable generating functionals. A similar result holds for random variables belonging to the domain of semistable attraction of (X_t). These investigations generalize results to the case of probabilities on groups which were recently obtained for vector spaces in [1]. Furthermore, distributions of such stable Lévy proceses are representable as limits of random products of semistable laws. [1] Becker-Kern, P., Scheffler, H-P.: How to find stability in a purely semistable context. Yokohama Math. J. 51, 75-88 (2005)