A Gaussian correlation inequality and its applications to the existence of small ball constant

Stochastic Processes and their Applications

Volumn 107, Issue 2, 2003, Pages 269-287

  Shao, Qi Man ^a,b  

^a UNIVERSITY OF OREGON   (United States)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Fraction Brownian motion; Gaussian correlation conjecture; Small ball problem

Indexed keywords

EID: 0041928037     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(03)00084-X     Document Type: Article

References (25)

1. Anderson T.W. The integral of a symmetric unimodal function over a symmetric set and some probability inequalities Proc. Amer. Math. Soc. 6 1955 170-176
2. Bapat R.B. Raghavan T.E.S. Nonnegative Matrices and Applications 1996 Cambridge University Press Cambridge
3. Beckenbach E. Bellman R. Inequalities 1965 Springer New York
4. Hargé G. A particular case of correlation inequality for Gaussian measure Ann. Probab. 27 1999 1939-1951
5. Hitczenko P. Kwapien S. Li W.V. Schechtman G. Schlumprecht T. Zinn J. Hypercontractivity and comparison of moments of iterated maxima and minima of independent random variables Electron. J. Probab. 3 2 1998 1-26
6. Hu Y. Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities J. Theoret. Probab. 10 1997 835-848
7. Khatri C. On certain inequalities for normal distributions and their applications to simultaneous confidence bounds Ann. Math. Statist. 38 1967 1853-1867
8. Koldobsky A. Montgomery-Smith S.J. Inequalities of correlation type for symmetric stable random vectors Statist. Probab. Lett. 28 1996 91-97
9. Kuelbs J. Li W.V. Metric entropy and the small ball problem for Gaussian measures J. Funct. Anal. 116 1993 133-157
10. Ledoux, M., 1996. Isoperimetry and Gaussian Analysis. Lecture Notes in Mathematics, Vol. 1648, Springer, Berlin, pp. 165-294.
11. Li W.V. Linde W. Existence of small ball constants for fractional Brownian motions C. R. Acad. Sci. Paris Sér. I Math. 326 1998 1329-1334
12. Li, W.V., Shao, Q.M., 2000. A note on the Gaussian correlation conjecture. In: Giné, E., Mason, D.M., Wellner, J.A. (Eds.), Progress in Probability, Vol. 47, Birkhéuser, Boston, pp. 163-172.
13. Monrad D. Rootzén H. Small values of Gaussian processes and functional laws of the iterated logarithm Probab. Theory Related Fields 101 1995 173-192
14. Pitt L.D. A Gaussian correlation inequality for symmetric convex sets Ann. Probab. 5 1977 470-474
15. Pitt L.D. Local times for Gaussian vector fields Indiana Univ. J. Math. 27 1978 309-330
16. Schechtman G. Schlumprecht T. Zinn J. On the Gaussian measure of the intersection Ann. Probab. 26 1998 346-357
17. Sidák Z. On multivariate normal probabilities of rectangles, their dependence on correlations Ann. Math. Statist. 39 1968 1425-1434
18. Shao Q.M. A note on small ball probability of Gaussian processes with stationary increments J. Theoret. Probab. 6 1993 595-602
19. Shao Q.M. Bounds and estimates of a basic constant in extreme value theory of Gaussian processes Statist. Sinica 6 1996 245-257
20. Shao Q.M. Wang D. Small ball probabilities of Gaussian fields Probab. Theory Related Fields 102 1995 511-517
21. Szarek S. Werner E. A nonsymmetric correlation inequality for Gaussian measure J. Multivariate Anal. 68 1999 193-211
22. Talagrand M. New Gaussian estimates for enlarged balls Geom. Funct. Anal. 3 1993 502-526
23. Talagrand M. The small ball problem for the Brownian sheet Ann. Probab. 22 1994 1331-1354
24. Talagrand M. Lower classes for fractional Brownian motion J. Theoret. Probab. 9 1996 191-213
25. Tong Y.L. Probability Inequalities in Multivariate Distributions 1980 Academic Press New York