Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order
Yihan Zhang
math.PR cs.IT math.ST stat.ML
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| $.
Topics
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