The Annals of Probability

The Wills functional and Gaussian processes

Richard A. Vitale

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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

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

First available in Project Euclid: 6 January 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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