Geodesic
Small ball probabilities for stable convolutions
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 327-343

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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 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.

DOI : 10.1051/ps:2007022
Classification : 60F99, 60G15, 60G20, 60G52
Keywords: entropy numbers, fractional Ornstein-Uhlenbeck processes, Riemann-Liouville processes, small ball probabilities, stochastic convolutions, wavelets 
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     author = {Aurzada, Frank and Simon, Thomas},
     title = {Small ball probabilities for stable convolutions},
     journal = {ESAIM: Probability and Statistics},
     pages = {327--343},
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     year = {2007},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2007022/}
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Aurzada, Frank; Simon, Thomas. Small ball probabilities for stable convolutions. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 327-343. doi: 10.1051/ps:2007022

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