Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

Yihan Zhang

arxiv: 2604.26819 · v1 · submitted 2026-04-29 · 🧮 math.PR · cs.IT· math.IT· math.ST· stat.ML· stat.TH

Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

Yihan Zhang This is my paper

Pith reviewed 2026-05-07 12:11 UTC · model grok-4.3

classification 🧮 math.PR cs.ITmath.ITmath.STstat.MLstat.TH

keywords mathbbconvexorderattainedboundedcomparisondominatedequality

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The pith

Any sub-Gaussian random variable (MGF bounded by standard normal) is dominated in convex order by G/E[|G|] where G is standard normal, with equality for the Rademacher distribution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

A random variable is sub-Gaussian if its moment generating function, which tracks its growth via exponentials, stays below that of a standard bell-curve normal. The result shows that for any function that bends upwards (convex), the average value of that function applied to the sub-Gaussian variable is at most the average for a slightly stretched normal. The stretch factor is one divided by the normal's average absolute size. The bound is tight: it holds with equality when the variable only takes values plus or minus one equally often, and the function is absolute value.

Core claim

any random variable X whose moment generating function is point-wise upper bounded by that of G ~ N(0,1) must be dominated by G/E[|G|] in convex order, meaning E[f(X)] ≤ E[f(G/E[|G|])] for all convex f. Equality is attained by X ~ Unif({-1,1}) and f(x)=|x|.

Load-bearing premise

The pointwise MGF upper bound by the standard normal's MGF holds for the random variable X, together with the one-dimensional setting and standard properties of convex order.

read the 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| $.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard definitions of moment generating functions and convex order; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)

standard math Standard properties of moment generating functions and convex stochastic order in one dimension
Invoked implicitly to connect the MGF bound to the convex-order domination.

pith-pipeline@v0.9.0 · 5392 in / 1174 out tokens · 77676 ms · 2026-05-07T12:11:25.153115+00:00 · methodology

Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

Core claim

Load-bearing premise

discussion (0)