Moment Inequalities for Suprema of Gaussian Random Processes

Abstract

Suppose $(X_t{)}_{t\in T}$ is a Gaussian process indexed by some arbitrary set $T:$ the random variable ${\sup }_{t\in T}{X}_t$ can be very intricate and bounding its expectation is a natural step towards understanding it. Sudakov-Fernique inequality allows to order expectations of suprema of such random processes: if $(X_t{)}_{t\in T},(Y_t{)}_{t\in T}$ are centered Gaussian random processes satisfying $\mathbb{E}[(X_t-X_s{)}^2]\le \mathbb{E}[(Y_t-Y_s{)}^2]$ for all $t,s\in T,$ then $\mathbb{E}[{\sup }_{t\in T}{X}_t]\le \mathbb{E}[{\sup }_{t\in T}{Y}_t].$ This work obtains similar results for higher moments under a slightly stronger condition than the one aforementioned.