Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

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 $f(x) = |x|$.

No reviews yet.

No comments yet.