Admissible Sequences for Talagrand's $\gamma_2$-functional

TL;DR

This paper investigates the $oldsymbol{ ext{γ}_2}$-functional in Gaussian processes, aiming to construct nearly optimal admissible sequences to better understand the expectations of process suprema.

Contribution

It introduces methods to build admissible sequences close to optimal, enhancing the use of the $oldsymbol{ ext{γ}_2}$-functional as a proxy for process supremum expectations.

Findings

Constructed admissible sequences that approximate optimality.

Provided insights into the $oldsymbol{ ext{γ}_2}$-functional's role in Gaussian process analysis.

Enhanced understanding of the expectation of suprema in Gaussian processes.

Abstract

Suprema of random processes appear naturally in a plethora of disciplines, and Talagrand's majorizing theorem yields a geometric interpretation for them: for a centered Gaussian random process $(X_t)_{t \in T},$ $\mathbb{E}[\sup_{t \in T}{X_t}]$ is comparable to the $\gamma_2$-functional of $T,$ a quantity that depends solely on the space $(T,d),$ where $d$ denotes the pseudometric $d(u,v)=\sqrt{\mathbb{E}[(X_u-X_v)^2]}.$ Despite the explicit definition of this functional, an infimum over admissible sequences, this tool tends to be used exclusively as a means to bound the expectation of the supremum of a random process by that of another. This work considers the $\gamma_2$-functional as a proxy for the quantity of interest by constructing admissible sequences that are close to being optimal, and aims to provide a promising avenue towards understanding expectations of suprema of Gaussian random processes.