Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order
Provides a sharp theoretical bound in probability theory for comparing sub-Gaussian random variables in convex order, relevant to researchers in probability and statistics.
The paper proves that any random variable with moment generating function bounded by that of a standard Gaussian is dominated in convex order by a scaled Gaussian, with the bound being sharp as shown by the Rademacher distribution.
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| $.