Small ball probabilities for stable convolutions

¹ Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany; aurzada@math.tu-berlin.de
² Equipe d'analyse et probabilités, Université d'Evry-Val d'Essonne, boulevard François Mitterrand, 91025 Evry Cedex, France; tsimon@univ-evry.fr

Received: 4 April 2006
Revised: 10 July 2006

Abstract

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function with a real SαS Lévy process. We show that the small ball exponent is uniquely determined by the norm and by the behaviour of f at zero, which extends the results of Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 725–752 where this was proved for f being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 725–752 is obtained with respect to the norms, thanks to a weak decorrelation inequality due to Li, Elec. Comm. Probab. 4 (1999) 111–118. In the more difficult non-Gaussian case, we use a different method relying on comparison of entropy numbers and restrict ourselves to Hölder and L_p-norms.