The Annals of Probability

The Wills functional and Gaussian processes

Richard A. Vitale

Full-text: Open access

Abstract

The Wills functional from the theory of lattice point enumeration can be adapted to produce the following exponential inequality for zero-mean Gaussian processes: $$E \exp [\sup_t (X_t - (1/2) \sigma_t^2)] \leq \exp (E \sup_t X_t).$$

An application is a new proof of the deviation inequality for the supremum of a Gaussian process above its mean:

$$P(\sup_t X_t - E \sup_t X_t \geq a) \leq \exp (-\frac{(1/2) \alpha^2}{\sigma^2}),$$

where $a > 0$ and $\sigma^2 = \sup_t \sigma_t^2$.

Article information

Source
Ann. Probab., Volume 24, Number 4 (1996), 2172-2178.

Dates
First available in Project Euclid: 6 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1041903224

Digital Object Identifier
doi:10.1214/aop/1041903224

Mathematical Reviews number (MathSciNet)
MR1415247

Zentralblatt MATH identifier
0879.60036

Subjects
Primary: 60G15: Gaussian processes
Secondary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 60G17: Sample path properties

Keywords
Alexandrov-Fenchel inequality Gaussian process deviation inequality exponential bound intrinsic volume mixed volume quermassintegral tail bound Wills functional

Citation

Vitale, Richard A. The Wills functional and Gaussian processes. Ann. Probab. 24 (1996), no. 4, 2172--2178. doi:10.1214/aop/1041903224. https://projecteuclid.org/euclid.aop/1041903224


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References

  • 1 BADRIKIAN, A. and CHEVET, S. 1974. Mesures cy lindriques, espaces de Wiener et fonctions aleatoires gaussiennes. Lecture Notes in Math. 379. Springer, New York. ´
  • 2 BURAGO, YU. D. and ZALGALLER, V. A. 1988. Geometric Inequalities. Springer, New York.
  • 3 CHEVET, S. 1976. Processus Gaussiens et volumes mixtes. Z. Wahrsch. Verw. Gebiete 36 47 65.
  • 6 FERNIQUE, X. 1995. Fonctions aleatoires gaussiennes, vecteurs aleatoires gaussiens troi´ ´. sieme version. Unpublished manuscript.
  • 7 GRITZMANN, P. and WILLS, J. M. 1993. Lattice points. In Handbook of Convex Geometry Z. P. M. Gruber and J. M. Wills, eds. B 765 797. North-Holland, Amsterdam.
  • 9 LEDOUX, M. and TALAGRAND, M. 1991. Probability in Banach Spaces. Springer, New York.
  • 10 MCMULLEN, P. 1975. Non-linear angle sum relations for poly hedral cones and poly topes. Math. Proc. Cambridge Philos. 78 247 260.
  • 11 MCMULLEN, P. 1991. Inequalities between intrinsic volumes. Monatsh. Math. 111 47 53.
  • 12 MILMAN, V. D. and PISIER, G. 1987. Gaussian processes and mixed volumes. Ann. Probab. 15 292 304.
  • 13 PISIER, G. 1986. Probabilistic methods in the geometry of Banach spaces. Probability and Analy sis. Lecture Notes in Math. 1206 167 241. Springer, New York.
  • 14 PISIER, G. 1989. The Volume of Convex Bodies and Banach Space Geometry. Springer, New York.
  • 15 SCHNEIDER, R. 1993. Convex Bodies: The Brunn Minkowski Theory. Cambridge Univ. Press.
  • 16 SUDAKOV, V. N. 1979. Geometric Problems in the Theory of Infinite-Dimensional Probability Distributions. Proceedings of the Steklov Institute of Mathematics No. 2. Amer. Math. Soc., Providence, RI.
  • 20 VITALE, R. A. 1988. An alternate formulation of mean value for random geometric figures. J. Microscopy 151 197 204.
  • 21 VITALE, R. A. 1993. A class of bounds for convex bodies in Hilbert space. Set-Valued Anal. 1 89 96.
  • 22 VITALE, R. A. 1995. On the volume of parallel bodies: a probabilistic derivation of the Steiner formula. Adv. in Appl. Probab. 27 97 101.
  • 23 WILLS, J. M. 1973. Zur Gitterpunktanzahl konvexer Mengen. Elem. Math. 28 57 63.
  • STORRS, CONNECTICUT 06269 E-MAIL: rvitale@uconnum.uconn.edu