Abstract
A sharp small ball estimate under Sobolev type norms is obtained for certain Gaussian processes in general and for fractional Brownian motions in particular. New method using the techniques in large deviation theory is developed for small ball estimates. As an application the Chung's LIL for fractional Brownian motions is given in this setting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Airault, H., and Malliavin, P. (1988). Intégration géométrique sur l'espace de Wiener. Bull. Sci. Math. 112, 3–52.
Baldi, P., and Roynette, B. (1992). Some exact equivalents for the Brownian motion in Hölder norm. Prob. Th. Rel. Fields 93, 457–484.
Borovkov, A. A., and Mogulskii, A. A. (1991). On probabilities of small deviations for stochastic processes. Siberian Adv. Math. 1, 39–63.
Csörgő, M., Horváth, L., and Shao, Q. M. (1993). Convergence of integrals of uniform empirical and quantile processes. Stoch. Process. 45, 283–294.
Csörgő, M., and Shao, Q. M. (1994). On almost sure limit inferior for B-valued stochastic processes and applications. Prob. Th. Rel. Fields 99, 29–54.
de Acosta, A. (1983). Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Prob. 11, 78–101.
Fang, S. (1991a). Sur un résultat d'entropie de l'espace de Wiener. C.R. Acad. Sci. Paris, t. Série I. 312, 995–998.
Fang, S. (1991b). Non-dégńérescence des pseudo-norms de Sobolev sur l'espace de Wiener. Bull. Sci. Math. 115, 223–234.
Fang, S. (1992). Quelque résultats d'entropie sur l'espace de Wiener. Potential Analysis 1, 167–176.
Fernique, X. (1964). Continuité des processus Gaussiens. Comptes Rendus 258, 6058–6060.
Garsia, A. (1976). Combinatorial inequalities and smoothness of functions. Bull. Amer. Math. Soc. 82, 157–170.
Jain, N. C., and Marcus, M. B. (1978). Continuity of sub-Gaussian processes. Advances in Probability, Vol. 4. Dekker, New York, pp. 81–196.
Khatri, C. G. (1967). On certain inequalities for normal distributions and their aplications to simultaneous confidence bounds. Ann. Math. Stat. 38, 1853–1867.
Kuelbs, J., and Li, W.V. (1992a). Metric entropy and the small ball problem for Gaussian measures. J. Functional Anal. 116, 133–157.
Kuelbs, J., and Li, W. V. (1992b). Small ball estimates for Brownian motion and the Brownian sheet. J. Theor. Prob. 6, 547–577.
Kuelbs, J., Li, W. V., and Shao, Q. M. (1995). Small ball probability for Gaussian processes with stationary increments under Hölder norms. J. Theoret. Prob. 8, 361–386.
Kuelbs, J., Li, W. V., and Talagrand, M. (1994). Lim inf results for Gaussian samples and Chung's functional LIL. Ann. Prob. 22, 1879–1903.
Marcus, M. B. (1968). Gaussian process with stationary increments possessing discontinuous sample paths. Pacific J. Math. 26, 149–157.
Monrad, D., and Rootzén, H. (1995). Small values of Gaussian processes and functional laws of iterated logarithm. Prob. Th. Rel. Fields 101, 173–192.
Shao, Q. M. (1993). A note on small ball probability of Gaussian process with stationary increments. J. Theor. Prob. 6, 595–602.
Shao, Q. M., and Wang, D. (1995). Small ball probabilities of Gaussian fields. Prob. Th. Rel. Fields 102, 511–517.
Sidak, Z. (1968). On multivariate normal probabilities of rectangles: Their dependence on correlations. Ann. Math. Stat. 39, 1425–1434.
Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501.
Stolz, W. (1993). Une méthods élémentaire pour l'évaluation de petites boules Browniennes. C.R. Acad. Sci. Paris 315, 1217–1220.
Talagrand, M. (1993). New Gaussian estimates for enlarged balls. Geom. Functional Anal. 3, 502–526.
Talagrand, M. (1994). The small ball problem for the Brownian sheet. Ann. Prob. 22, 1331–1354.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Li, W.V., Shao, QM. Small Ball Estimates for Gaussian Processes under Sobolev Type Norms. Journal of Theoretical Probability 12, 699–720 (1999). https://doi.org/10.1023/A:1021675731663
Issue Date:
DOI: https://doi.org/10.1023/A:1021675731663