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

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 : https://doi.org/10.1051/ps:2007022
Classification : 60F99,  60G15,  60G20,  60G52
Mots clés : 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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     language = {en},
     url = {http://www.numdam.org/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. http://www.numdam.org/articles/10.1051/ps:2007022/

[1] S. Artstein, V. Milman and S.J. Szarek, Duality of metric entropy. Ann. Math. (2) 159 (2004) 1213-1328. | Zbl 1072.52001

[2] F. Aurzada., Metric entropy and the small deviation problem for stable processes. To appear in Prob. Math. Stat. (2006). | Zbl 1136.60033

[3] P. Berthet and Z. Shi, Small ball estimates for Brownian motion under a weighted sup-norm. Studia Sci. Math. Hungar. 36 (2000) 275-289. | Zbl 0973.60083

[4] J. Bertoin, On the first exit time of a completely asymmetric stable process from a finite interval. Bull. London Math. Soc. 28 (1996) 514-520. | Zbl 0863.60068

[5] P. Cheridito, H. Kawaguchi and M. Maejima, Fractional Ornstein-Uhlenbeck Processes. Elec. J. Probab. 8 (2003) 1-14. | EuDML 124790 | Zbl 1065.60033

[6] D.E. Edmunds and H. Triebel, Function spaces, entropy numbers, differential operators. Cambridge University Press (1996). | MR 1410258 | Zbl 0865.46020

[7] G.H. Hardy and J.E. Littlewood, Some properties of fractional integrals I. Math. Z. 27 (1927) 565-606. | EuDML 167995 | JFM 54.0275.05

[8] D. Heath, R.A. Jarrow and A. Morton, Bond pricing and the term structure of interest rate: A new methodology for contingent claim valuation. Econometrica 60 (1992) 77-105. | Zbl 0751.90009

[9] J. Kuelbs and W.V. Li, Metric Entropy and the Small Ball Problem for Gaussian Measures. J. Func. Anal. 116 (1993) 113-157. | Zbl 0799.46053

[10] S. Kwapień, M.B. Marcus and J. Rosiński, Two results on continuity and boundedness of stochastic convolutions. Ann. Inst. Henri Poincaré, Probab. et Stat. 42 (2006) 553-566. | Numdam | Zbl 1106.60040

[11] W.V. Li, A Gaussian correlation inequality and its application to small ball probabilities. Elec. Comm. Probab. 4 (1999) 111-118. | Zbl 0937.60026

[12] W.V. Li, Small ball probabilities for Gaussian Markov processes under the L p -norm. Stochastic Processes Appl. 92 (2001) 87-102. | Zbl 1047.60031

[13] W.V. Li and W. Linde, Existence of small ball constants for fractional Brownian motions. C. R. Acad. Sci. Paris 326 (1998) 1329-1334. | Zbl 0922.60039

[14] W.V. Li and W. Linde, Approximation, metric entropy and the small ball problem for Gaussian measures. Ann. Probab. 27 (1999) 1556-1578. | Zbl 0983.60026

[15] W.V. Li and W. Linde, Small Deviations of Stable Processes via Metric Entropy. J. Theoret. Probab. 17 (2004) 261-284. | Zbl 1057.60047

[16] M.A. Lifshits and T. Simon, Small deviations for fractional stable processes. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 725-752. | Numdam | Zbl 1070.60042

[17] M. Maejima and K. Yamamoto, Long-Memory Stable Ornstein-Uhlenbeck Processes. Elec. J. Probab. 8 (2003) 1-18. | Zbl 1087.60034

[18] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach (1993). | MR 1347689 | Zbl 0818.26003

[19] G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes. Chapman & Hall (1994). | MR 1280932 | Zbl 0925.60027

[20] K. Takashima, Sample path properties of ergodic self-similar processes. Osaka J. Math. 26 (1989) 159-189. | Zbl 0719.60040

[21] M.S. Taqqu and R.L. Wolpert, Fractional Ornstein-Uhlenbeck Lévy Processes and the Telecom Process: Upstairs and Downstairs. Signal Processing 85 (2005) 1523-1545.

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