Small deviations for fractional stable processes

Lifshits, Mikhail; Simon, Thomas

doi:10.1016/j.anihpb.2004.05.004

Lifshits, Mikhail ; Simon, Thomas

Annales de l'I.H.P. Probabilités et statistiques, Tome 41 (2005) no. 4, pp. 725-752.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpb.2004.05.004

@article{AIHPB_2005__41_4_725_0,
     author = {Lifshits, Mikhail and Simon, Thomas},
     title = {Small deviations for fractional stable processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {725--752},
     publisher = {Elsevier},
     volume = {41},
     number = {4},
     year = {2005},
     doi = {10.1016/j.anihpb.2004.05.004},
     mrnumber = {2144231},
     zbl = {1070.60042},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpb.2004.05.004/}
}

TY  - JOUR
AU  - Lifshits, Mikhail
AU  - Simon, Thomas
TI  - Small deviations for fractional stable processes
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2005
SP  - 725
EP  - 752
VL  - 41
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpb.2004.05.004/
DO  - 10.1016/j.anihpb.2004.05.004
LA  - en
ID  - AIHPB_2005__41_4_725_0
ER  -

%0 Journal Article
%A Lifshits, Mikhail
%A Simon, Thomas
%T Small deviations for fractional stable processes
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2005
%P 725-752
%V 41
%N 4
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpb.2004.05.004/
%R 10.1016/j.anihpb.2004.05.004
%G en
%F AIHPB_2005__41_4_725_0

Lifshits, Mikhail; Simon, Thomas. Small deviations for fractional stable processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 41 (2005) no. 4, pp. 725-752. doi : 10.1016/j.anihpb.2004.05.004. http://archive.numdam.org/articles/10.1016/j.anihpb.2004.05.004/

Bibliographie
Cité par

[1] A. Ayache, M. Taqqu, Rate optimality of wavelet series approximations of fractional Brownian motion, J. Fourier Anal. Appl. 9 (5) (2003) 451-471. | MR | Zbl

[2] P. Baldi, B. Roynette, Some exact equivalents for the Brownian motion in Hölder semi-norm, Probab. Theory Related Fields 93 (4) (1992) 457-484. | MR | Zbl

[3] E. Belinsky, W. Linde, Small ball probabilities of fractional Brownian sheets via fractional integration operators, J. Theoret. Probab. 15 (3) (2002) 589-612. | MR | Zbl

[4] P. Berthet, Z. Shi, Small ball estimates for Brownian motion under a weighted sup-norm, Studia Sci. Math. Hung. 36 (1-2) (2001) 275-289. | MR | Zbl

[5] J. Bertoin, On the first exit time of a completely asymmetric stable process from a finite interval, Bull. London Math. Soc. 28 (5) (1996) 514-520. | MR | Zbl

[6] N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. | MR | Zbl

[7] R.M. Blumenthal, R.K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960) 263-273. | MR | Zbl

[8] A.A. Borovkov, A.A. Mogulskii, On probabilities of small deviations for stochastic processes, Siberian Adv. Math. 1 (1) (1991) 39-63. | MR | Zbl

[9] J.C. Bronski, Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions, J. Theoret. Probab. 16 (1) (2003) 87-100. | MR | Zbl

[10] X. Chen, J. Kuelbs, W.V. Li, A functional LIL for symmetric stable processes, Ann. Probab. 28 (1) (2000) 258-277. | MR | Zbl

[11] X. Chen, Quadratic functionals and small ball probabilities for the m-fold integrated Brownian motion, Ann. Probab. 31 (2) (2003) 1052-1077. | MR | Zbl

[12] V.V. Chistyakov, O.E. Galkin, On maps of bounded p-variation with $p > 1$ , Positivity 2 (1) (1998) 19-45. | MR | Zbl

[13] Z. Ciesielski, G. Kerkyacharian, B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math. 107 (2) (1993) 171-204. | EuDML | MR | Zbl

[14] M. Csörgö, E. Horváth, Q.-M. Shao, Convergence of integrals of uniform empirical and quantile processes, Stochastic Process. Appl. 45 (2) (1993) 283-294. | MR | Zbl

[15] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. | MR | Zbl

[16] C. Donati-Martin, S. Song, M. Yor, Symmetric stable processes, Fubini's theorem, and some extensions of the Ciesielski-Taylor identities in law, Stochastic Stochastic Rep. 50 (1-2) (1994) 1-33. | MR | Zbl

[17] R.M. Dudley, R. Norvaiša, An introduction to p-variation and Young integrals. With emphasis on sample functions of stochastic processes, MaPhySto Lect. Notes, vol. 1, 1998, Univ. of Aarhus. | Zbl

[18] P. Embrechts, M. Maejima, Self-similar Processes, Princeton University Press, Princeton, 2002. | Zbl

[19] D. Khoshnevisan, Z. Shi, Chung's law for integrated Brownian motion, Trans. Amer. Math. Soc. 350 (10) (1998) 4253-4264. | MR | Zbl

[20] N. Kôno, M. Maejima, Hölder continuity of sample paths of some self-similar stable processes, Tokyo J. Math. 14 (1) (1991) 93-100. | MR | Zbl

[21] J. Kuelbs, W.V. Li, Small ball problems for Brownian motion and for the Brownian sheet, J. Theoret. Probab. 6 (3) (1993) 547-577. | MR | Zbl

[22] J. Kuelbs, W.V. Li, Q.-M. Shao, Small ball probabilities for Gaussian processes with stationary increments under Hölder norms, J. Theoret. Probab. 8 (2) (1995) 361-386. | MR | Zbl

[23] W.V. Li, W. Linde, Existence of small ball constants for fractional Brownian motions, C. R. Acad. Sci. Paris 326 (11) (1998) 1329-1334. | MR | Zbl

[24] W.V. Li, W. Linde, Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab. 27 (3) (1999) 1556-1578. | MR | Zbl

[25] W.V. Li, Q.-M. Shao, Small ball estimates for Gaussian processes under Sobolev type norms, J. Theoret. Probab. 12 (3) (1999) 699-720. | MR | Zbl

[26] W.V. Li, Q.-M. Shao, Gaussian processes: inequalities, small ball probabilities and applications, in: Stochastic Processes: Theory and Methods, Handbook of Statistics, vol. 19, 2001, pp. 533-597. | MR | Zbl

[27] M.A. Lifshits, Asymptotic behavior of small ball probabilities, in: Probab. Theory and Math. Statist., Proc. VII International Vilnius Conference (1998), VSP/TEV, Vilnius, 1999, pp. 453-468. | Zbl

[28] M.A. Lifshits, W. Linde, Approximation and entropy numbers of Volterra operators with application to Brownian motion, Mem. Amer. Math. Soc. 745 (2002). | MR | Zbl

[29] M.A. Lifshits, W. Linde, Small deviations of weighted fractional processes and average non-linear approximation, Trans. Amer. Math. Soc., 2002, in press. | MR | Zbl

[30] D. Marinucci, P.M. Robinson, Alternative forms of fractional Brownian motion, J. Stat. Plann. Inference 80 (1-2) (1999) 111-122. | MR | Zbl

[31] Y. Meyer, Wavelets and Operators, Cambridge Studies in Advanced Mathematics, Cambridge, 1992. | MR | Zbl

[32] A.A. Mogulskii, Small deviations in a space of trajectories, Theor. Probab. Appl. 19 (1974) 726-736. | MR | Zbl

[33] B. Roynette, Mouvement brownien et espaces de Besov, Stochastic Stochastic Rep. 43 (3-4) (1993) 221-260. | MR | Zbl

[34] M. Ryznar, Asymptotic behavior of stable seminorms near the origin, Ann. Probab. 14 (1) (1986) 287-298. | MR | Zbl

[35] G. Samorodnitsky, Lower tails of self-similar stable processes, Bernoulli 4 (1) (1998) 127-142. | MR | Zbl

[36] G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, New York, 1994. | MR | Zbl

[37] Q.-M. Shao, A note on small ball probability of a Gaussian process with stationary increments, J. Theoret. Probab. 6 (3) (1993) 595-602. | MR | Zbl

[38] Q.-M. Shao, A Gaussian correlation inequality and its application to the existence of small ball constant, Stochastic Process. Appl. 107 (2) (2003) 269-287. | MR | Zbl

[39] Z. Shi, Lower tails of quadratic functionals of symmetric stable processes, Prépublication de l'Université Paris-VI, 1999.

[40] T. Simon, Small ball estimates in p-variations for stable processes, J. Theoret. Probab., 2003, in press. | MR | Zbl

[41] W. Stolz, Une méthode élémentaire pour l'évaluation des petites boules browniennes, C. R. Acad. Sci. Paris 316 (11) (1993) 1217-1220. | MR | Zbl

[42] W. Stolz, Small ball probabilities for Gaussian processes under non-uniform norms, J. Theoret. Probab. 9 (3) (1996) 613-630. | MR | Zbl

[43] K. Takashima, Sample path properties of ergodic self-similar processes, Osaka J. Math. 26 (1) (1989) 159-189. | MR | Zbl

[44] S.J. Taylor, Sample path properties of a transient stable process, J. Math. Mech. 16 (1967) 1229-1246. | MR | Zbl

Cité par Sources :