The Wills functional and Gaussian processes
Richard A. Vitale · The Annals of Probability · 1996
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 $\sigma^2 = \sup_t \sigma_t^2$.