Summary
Lower bounds on the small ball probability are given for Brownian sheet type Gaussian fields as well as for general Gaussian fields with stationary increments in ℝd. In particular, a sharp bound is found for the fractional Lévy Brownian fields.
References
Bass, R.F.: Probability estimates for multiparameter Brownian processes. Ann. Probab.16, 251–264 (1988)
Csörgő, M., Shao, Q.M.: On almost sure limit inferior for B-valued stochastic processes and applications. Probab. Theory Relat. Fields99, 29–54 (1994)
Fernique, X.: Regularité des trajectoires des fonctions aléatoires Gaussiennes (Lect. Notes Math., vol.480, pp. 1–96) Berlin: Springer 1975
Khatri, C.: On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist.38, 1853–1867 (1967)
Kôno, N.: Evolution asymptotique des temps d'arrêt et des temps de séjour liés aux trajectoires de certaines fonctions aléatoires gaussiennes. (Lecture Notes. Math., vol550, pp. 290–296) Berlin; Springer 1976
Kuelbs, J., Li, W.V.: Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal.116, 133–157 (1992)
Kuelbs, J., Li, W.V., Shao, Q.M.: Small ball probabilities for Gaussian processes with stationary increments under Hölder norms. J. Theoret. Probab. (to appear) (1995)
Ledoux, M.: Isoperimetry and Gaussian analysis (manuscript, 1994)
Li, W.V., Shao, Q.M.: Small ball probabilities for Gaussian processes under Sobolev type norms (submitted)
Monrad, D., Rootzén, H.: Small values of fractional Brownian motion and locally nondeterministic, Gaussian processes (manuscript, 1992)
Šidák, Z.: On multivariate normal probabilities of rectangles, their dependence on correlations. Ann. Math. Statist.39, 1425–1434 (1968)
Shao, Q.M.: A note on small ball probability of Gaussian processes with stationary increments. J. Theoret. Probab.6, 595–602 (1993)
Shao, Q.M.: Bounds and estimates of a basic constant in extreme value theory of Gaussian processes. Statist. Sinica (to appear)
Shao, Q.M., Wang, D.: Small ball probabilities of Gaussian fields. Dept. of Math., National Univ. of Singapore, Research Report No. 616 (1994)
Talagrand, M.: New Gaussian estimates for enlarged balls. Geometric Funct. Anal.3, 502–526 (1993)
Talagrand, M.: The small ball problem for the Brownian sheet. Ann. Probab.22, 1331–1354 (1994)
Talagrand, M.: Hausdorf measure of trajectories of multiparameter fractional Brownian motion (manuscript, 1994b)
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The research is partly supported by a National University of Singapore's Research Project
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Shao, Q.M., Wang, D. Small ball probabilities of Gaussian fields. Probab. Th. Rel. Fields 102, 511–517 (1995). https://doi.org/10.1007/BF01198847
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DOI: https://doi.org/10.1007/BF01198847
Mathematics Subject Classification (1991)
- 60G15
- 60G18
- 60F15