Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

TL;DR

This paper establishes a sharp comparison between sub-Gaussian random variables and a scaled Gaussian in convex order, providing a precise inequality for their expectations under convex functions.

Contribution

It proves that any sub-Gaussian variable with a bounded moment generating function is dominated by a scaled Gaussian in convex order, with equality cases characterized.

Findings

Sub-Gaussian variables are dominated by scaled Gaussian in convex order.

Equality occurs for specific distributions like the uniform distribution on {-1,1}.

The result provides a sharp comparison tool in probability theory.

Abstract

We prove that any random variable $X$ whose moment generating function is point-wise upper bounded by that of $ G \sim \mathcal{N}(0,1) $ must be dominated by $ G/\mathbb{E}[|G|] $ in convex order, meaning $ \mathbb{E}[f(X)] \le \mathbb{E}[f(G/\mathbb{E}[|G|])] $ for all convex $f$. Equality is attained by taking $ X \sim \mathrm{Unif}(\{-1,1\}) $ and $ f(x) = |x| $.