## Small ball probabilities for stable convolutions.(English)Zbl 1184.60010

Summary: 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 $$S\alpha S$$ 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 S. G. Lifshits and M. Simon [Ann. Inst. H. Poincaré Probab. Statist. 41, No. 4, 725–752 (2005; Zbl 1070.60042)] where this was proved for $$f$$ being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as Lifshits and Simon [loc. cit.], is obtained with respect to the norms, thanks to a weak decorrelation inequality due to Li [Elec. Comm. Probab. 4, 111–118 (1999; Zbl 0937.60026)]. 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.

### MSC:

 60G15 Gaussian processes 60F99 Limit theorems in probability theory 60G20 Generalized stochastic processes 60G52 Stable stochastic processes

### Citations:

Zbl 1070.60042; Zbl 0937.60026
Full Text:

### References:

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