Small ball probabilities for stable convolutions

Frank Aurzada; Thomas Simon

ESAIM: Probability and Statistics (2007)

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

Abstract

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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 Lp-norms.

How to cite

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Aurzada, 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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