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