The Wills functional and Gaussian processes

Vitale, Richard A.

Ann. Probab., Tome 24 (1996) no. 2, p. 2172-2178 / Harvested from Project Euclid

Résumé

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

Détail

Publié le : 1996-10-14
Classification:  Alexandrov-Fenchel inequality,  Gaussian process,  deviation inequality,  exponential bound,  intrinsic volume,  mixed volume,  quermassintegral,  tail bound,  Wills functional,  60G15,  52A20,  60G17