On the subgaussian comparison theorem

TL;DR

This paper proves that 1-subgaussian vectors are dominated by scaled Gaussian vectors in convex order, strengthening Talagrand's comparison theorem through a novel proof combining tensorization and classical ideas.

Contribution

It provides a strengthened version of Talagrand's subgaussian comparison theorem with a new proof technique.

Findings

1-subgaussian vectors are dominated by scaled Gaussian vectors in convex order

The proof combines tensorization with classical Fernique ideas

Strengthens existing subgaussian comparison results

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.