Small ball probabilities for stable convolutions

Aurzada, Frank; Simon, Thomas

ESAIM: Probability and Statistics, Tome 11 (2007), pp. 327-343.

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function $f:]0,+\infty [\to ℝ$ with a real $S\alpha S$ 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.

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