Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order
TLDR
This paper proves a sharp one-dimensional sub-Gaussian comparison, showing variables with bounded MGFs are dominated by a scaled Gaussian in convex order.
Key contributions
- Establishes a sharp 1D sub-Gaussian comparison result in convex order.
- Shows random variables with MGFs bounded by a standard Gaussian are dominated in convex order.
- The dominating variable is a specific scaled standard Gaussian, G/E[|G|].
- Demonstrates equality is attained by a uniform random variable X ~ Unif({-1,1}).
Why it matters
This paper offers a fundamental theoretical result in probability theory, providing a sharp sub-Gaussian comparison in convex order. This precise bound is crucial for applications in statistics, machine learning, and theoretical computer science that rely on sub-Gaussian properties. Its sharpness sets a strong benchmark for future theoretical work.
Original 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| $.
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