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Small ball probabilities for stable convolutions

Published online by Cambridge University Press:  17 August 2007

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

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function $f : \; ]0, +\infty[ \;\to \mathbb{R}$ 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) 725–752where 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.

Keywords

Entropy numbersfractional Ornstein-Uhlenbeck processesRiemann-Liouville processessmall ball probabilitiesstochastic convolutionswavelets.

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