Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

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 $f$. This is sharp as witnessed by $X\sim \mathrm{Unif}(\{-1,1\})$ and $f(x)=|x|$.