Small ball probabilities for stable convolutions

Abstract

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function f: ]0, +∞[→ R with a real SαS Levy 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. Poincare 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. Poincare 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 Holder and L p -norms.