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Moment Inequalities for Suprema of Gaussian Random Processes

Simona Diaconu

TL;DR

The paper tackles bounding higher moments of the supremum of Gaussian processes by extending the classical Sudakov-Fernique comparison to moments. It develops a Gaussian-interpolation framework with smooth max-approximants to convert covariance-difference bounds into moment inequalities, culminating in a main result that E[(max_i |X_i|)^m] ≤ E[(max_i |Y_i|)^m] under a strengthened condition, plus a corollary that yields explicit, computable bounds via a Gaussian shift. Auxiliary lemmas on covariance structure and correlation bounds underpin the proof and ensure control of error terms in the interpolation. The findings provide practical tools for bounding higher-order behavior of Gaussian maxima, with potential implications for tail estimates and concentration phenomena in Gaussian settings.

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.

Moment Inequalities for Suprema of Gaussian Random Processes

TL;DR

The paper tackles bounding higher moments of the supremum of Gaussian processes by extending the classical Sudakov-Fernique comparison to moments. It develops a Gaussian-interpolation framework with smooth max-approximants to convert covariance-difference bounds into moment inequalities, culminating in a main result that E[(max_i |X_i|)^m] ≤ E[(max_i |Y_i|)^m] under a strengthened condition, plus a corollary that yields explicit, computable bounds via a Gaussian shift. Auxiliary lemmas on covariance structure and correlation bounds underpin the proof and ensure control of error terms in the interpolation. The findings provide practical tools for bounding higher-order behavior of Gaussian maxima, with potential implications for tail estimates and concentration phenomena in Gaussian settings.

Abstract

Suppose is a Gaussian process indexed by some arbitrary set the random variable 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 are centered Gaussian random processes satisfying for all then This work obtains similar results for higher moments under a slightly stronger condition than the one aforementioned.

Paper Structure

This paper contains 4 sections, 5 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Monotonicity via Gaussian Intrapolation
  3. Proof of Theorem \ref{['extsudfer']}
  4. Auxiliary Lemmas

Key Result

Lemma 1

Suppose $X,Y \in \mathbb{R}^d$ are independent random vectorsBy a slight abuse of notation, any zero vector is denoted by $0.$ with $X\overset{d}{=}N(0,\Sigma^X),Y\overset{d}{=}N(0,\Sigma^Y),$ and let Then for any twice differentiable function $f:\mathbb{R}^d \to \mathbb{R},$ assuming all expectations involved are finite.

Theorems & Definitions (9)

  • Lemma 1: Gaussian Interpolation, Vershynin vershynin
  • Theorem 1
  • Corollary
  • proof
  • Remark
  • Lemma 2
  • proof
  • Lemma 3
  • proof