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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
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