# Small ball probabilities for stable convolutions

ESAIM: Probability and Statistics (2007)

- Volume: 11, page 327-343
- ISSN: 1292-8100

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topAurzada, Frank, and Simon, Thomas. "Small ball probabilities for stable convolutions." ESAIM: Probability and Statistics 11 (2007): 327-343. <http://eudml.org/doc/250118>.

@article{Aurzada2007,

abstract = {
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 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 Lp-norms.
},

author = {Aurzada, Frank, Simon, Thomas},

journal = {ESAIM: Probability and Statistics},

keywords = {Entropy numbers; fractional Ornstein-Uhlenbeck processes; Riemann-Liouville processes; small ball probabilities; stochastic convolutions; wavelets.; entropy numbers; riemann-liouville processes; wavelets},

language = {eng},

month = {8},

pages = {327-343},

publisher = {EDP Sciences},

title = {Small ball probabilities for stable convolutions},

url = {http://eudml.org/doc/250118},

volume = {11},

year = {2007},

}

TY - JOUR

AU - Aurzada, Frank

AU - Simon, Thomas

TI - Small ball probabilities for stable convolutions

JO - ESAIM: Probability and Statistics

DA - 2007/8//

PB - EDP Sciences

VL - 11

SP - 327

EP - 343

AB -
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 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 Lp-norms.

LA - eng

KW - Entropy numbers; fractional Ornstein-Uhlenbeck processes; Riemann-Liouville processes; small ball probabilities; stochastic convolutions; wavelets.; entropy numbers; riemann-liouville processes; wavelets

UR - http://eudml.org/doc/250118

ER -

## References

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