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 $\mathbb{E}[(X_t-X_s)^2] \leq \mathbb{E}[(Y_t-Y_s)^2]$ for all $t,s \in T,$ then $\mathbb{E}[\sup_{t \in T}{X_t}] \leq \mathbb{E}[\sup_{t \in T}{Y_t}].$ This work obtains similar results for higher moments under a slightly stronger condition than the one aforementioned.