Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

arXiv:2604.26819v1 Announce Type: cross 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| $.