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On the subgaussian comparison theorem

Ramon van Handel

TL;DR

The paper proves that any $1$-subgaussian vector is convexly dominated by a universal constant times a standard Gaussian, strengthening Talagrand's subgaussian comparison principle.It combines Fernique's functional framework with J. Liu's tensorization principle to reduce the problem to the study of stationary auxiliary processes and the classical majorizing-measure machinery.This yields a non-centered version of the majorizing measure inequality and provides an elementary probabilistic route to understanding suprema of general subgaussian processes, bypassing some of the geometric complexity of prior proofs.The approach also implies a Strassen-type coupling and connects to classical bounds for Gaussian suprema (Dudley–Fernique) within a broader convex-analytic context.

Abstract

The aim of this expository note is to prove that any $1$-subgaussian random vector is dominated in the convex ordering by a universal constant times a standard Gaussian vector. This strengthens Talagrand's celebrated subgaussian comparison theorem. The proof combines a tensorization argument due to J. Liu with ideas that date back to the work of Fernique.

On the subgaussian comparison theorem

TL;DR

The paper proves that any $1$-subgaussian vector is convexly dominated by a universal constant times a standard Gaussian, strengthening Talagrand's subgaussian comparison principle.It combines Fernique's functional framework with J. Liu's tensorization principle to reduce the problem to the study of stationary auxiliary processes and the classical majorizing-measure machinery.This yields a non-centered version of the majorizing measure inequality and provides an elementary probabilistic route to understanding suprema of general subgaussian processes, bypassing some of the geometric complexity of prior proofs.The approach also implies a Strassen-type coupling and connects to classical bounds for Gaussian suprema (Dudley–Fernique) within a broader convex-analytic context.

Abstract

The aim of this expository note is to prove that any -subgaussian random vector is dominated in the convex ordering by a universal constant times a standard Gaussian vector. This strengthens Talagrand's celebrated subgaussian comparison theorem. The proof combines a tensorization argument due to J. Liu with ideas that date back to the work of Fernique.

Paper Structure

This paper contains 10 sections, 8 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Random processes
  3. Fernique's functional
  4. Organization of this note
  5. On the work of Fernique
  6. Liu's tensorization principle
  7. Statement of the principle
  8. Two simple lemmas
  9. Proof of the tensorization principle
  10. Proof of Theorem \ref{['thm:mainprocess']}

Key Result

Theorem 1.1

Let $X$ be any $1$-subgaussian random vector in $\mathbb{R}^n$ and $G\sim N(0,I_n)$ be a standard Gaussian vector in $\mathbb{R}^n$. Then for every convex function $f:\mathbb{R}^n\to\mathbb{R}$, where $c$ is a universal constant.As every convex function is lower bounded by an affine function, the expectations $\mathbf{E}[f(X)]$ and $\mathbf{E}[f(cG)]$ are well defined for every convex function $f

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 2.1: Dudley; Fernique
  • Proposition 3.1: Liu's tensorization principle
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 4 more