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October 1996 The Wills functional and Gaussian processes
Richard A. Vitale
Ann. Probab. 24(4): 2172-2178 (October 1996). DOI: 10.1214/aop/1041903224

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 $\sigma^2 = \sup_t \sigma_t^2$.

Citation

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Richard A. Vitale. "The Wills functional and Gaussian processes." Ann. Probab. 24 (4) 2172 - 2178, October 1996. https://doi.org/10.1214/aop/1041903224

Information

Published: October 1996
First available in Project Euclid: 6 January 2003

zbMATH: 0879.60036
MathSciNet: MR1415247
Digital Object Identifier: 10.1214/aop/1041903224

Subjects:
Primary: 60G15
Secondary: 52A20 , 60G17

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

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 4 • October 1996
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