Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

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| $.

• The paper demonstrates that the optimal constant for convex order domination of MGF-sub-Gaussian variables is exactly √(π/2).
• It employs a hinge decomposition and reduces the analysis to two-point mixtures to rigorously compare with Gaussian distributions.
• The findings reveal a clear gap between MGF-based and tail-based sub-Gaussian comparisons, impacting risk quantification and concentration inequalities.

Sharp Convex Order Comparison for Sub-Gaussian Random Variables

Overview

The paper "Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order" (2604.26819) provides a precise characterization of the minimal scaling constant $c$ such that every real-valued, mean-zero random variable $X$ whose moment generating function (MGF) is upper bounded by that of a standard normal $G \sim \mathcal{N}(0,1)$ is dominated in convex order by $c G$. The author establishes that the optimal constant $c$ is exactly $\sqrt{\pi/2}$, thereby resolving an open problem regarding the relationship between sub-Gaussianity (in the MGF sense) and convex order comparison to Gaussian distributions.

Context, Motivation, and Problem Setting

Convex order comparison is a key tool in probability, allowing for sharp risk and uncertainty quantification that goes beyond simple second-moment (variance) domination. For $X, Y$ integrable, $X \leq_{cx} Y$ if $\mathbb{E} f(X) \leq \mathbb{E} f(Y)$ for every convex $f$ such that the expectations exist. Recent work (see van Handel et al.) showed the existence (but not minimality) of a universal $X$0 such that any centered, sub-Gaussian random vector $X$1 satisfies $X$2 where $X$3 is standard Gaussian. This immediately raises sharp formulation/optimization questions: for scalar random variables and in the MGF-based definition of sub-Gaussianity, what is the smallest $X$4 that works for all such $X$5?

The MGF-based sub-Gaussianity (hereafter, MGF-sub-Gaussian) considered is: for all $X$6, $X$7, where $X$8. The comparison constant

$X$9

is the central object of study. Prior literature [Davis, Power] addressed the optimal $G \sim \mathcal{N}(0,1)$0 for the tail sense version; the present analysis precisely targets the MGF-based formulation, revealing that the two are not equivalent and yield quantitatively different constants.

Main Results and Technical Contributions

The principal finding is the exact value:

$G \sim \mathcal{N}(0,1)$1

This is strictly less than the corresponding "tail sense" constant $G \sim \mathcal{N}(0,1)$2 from [Davis, Power], demonstrating that the MGF-based condition is quantitatively tighter. The author proves that for any $G \sim \mathcal{N}(0,1)$3 satisfying the MGF-sub-Gaussian condition, $G \sim \mathcal{N}(0,1)$4.

A crucial insight is the reduction, via the "hinge" decomposition of convex functions, that suffices to verify the desired inequality for two-point mixtures—specifically, symmetric Bernoulli random variables. The hinge representation $G \sim \mathcal{N}(0,1)$5 (with $G \sim \mathcal{N}(0,1)$6 constants and $G \sim \mathcal{N}(0,1)$7 a nonnegative measure) is leveraged to reduce the complexity of the convex order verification.

The sharpness of $G \sim \mathcal{N}(0,1)$8 is exhibited by $G \sim \mathcal{N}(0,1)$9 and $c G$0, where the exact threshold is attained:

$c G$1

which yields $c G$2.

Key Lemmas and Proof Structure

1. Reduction to Two-Point Laws: By conditioning on the event $c G$3 and matching first moments and MGFs, the analysis shows it suffices to compare with mixtures of Dirac masses at $c G$4 determined by the probability mass at each side of a threshold $c G$5.
2. Use of Hinge Functions: Convex order is checked via comparison of expectations of $c G$6 for all $c G$7, as these span all convex functions due to the hinge representation.
3. Explicit Gaussian Calculations: The value $c G$8 emerges as $c G$9 due to the normalization necessary to match the first absolute moment functional of Bernoulli random variables with the Gaussian.
4. Technical Bound on the Gaussian Isoperimetric Function: A lower bound $c$0 is established for the Gaussian isoperimetric function, necessary for the tail estimates in the proof, and the sharpness of a constant (see Lemma 1 and its appendices) is verified to within a uniform $c$1 gap.

Distinctions from Prior Art

The result demonstrates that the optimal comparison in convex order for MGF-sub-Gaussian random variables yields a constant that is markedly smaller than for the previously considered "tail" notion, emphasizing sensitivity to the precise operational definition of sub-Gaussianity. Furthermore, the sharp value of $c$2 arises from matching the extremal case for Rademacher random variables, contrasting with $c$3 where the extremal law is different.

This demonstrates that the conversion between MGF and tail sense sub-Gaussianity is lossy. The argument synthesizes ideas from convex analysis, probabilistic inequalities, and properties of the Gaussian law, producing an interpretable scaling in terms of standard Gaussian constants.

Limitations and Open Directions

The methods crucially rely on univariate properties, particularly the hinge decomposition of convex functions, and do not appear to generalize in a straightforward way to $c$4 for $c$5. As such, identifying the sharp constant in the multivariate setting for either the MGF or tail notion remains open. The lack of an analogous sharp representation of convex functions in higher dimensions is a significant barrier to such generalization.

The proof also does not yield new geometric or transportation-theoretic insights into the structure of the Gaussian isoperimetry or convex ordering, relying on analytic (rather than geometric) lower bounds on the isoperimetric function.

Implications and Future Research

This work sharpens the toolkit for comparing general sub-Gaussian random variables to Gaussian benchmarks in risk-sensitive and convex-functional contexts. This has implications for probabilistic inequalities, optimal transport, information theory, and high-dimensional statistics, especially for designing extremal bounds and calibrating concentration inequalities. Precisely identifying how extremality propagates through convex functional comparisons helps clarify the scope and limits of sub-Gaussian approximations.

There is potential for future work at several levels:

• Extension to vector-valued and infinite-dimensional random variables.
• Exploration of geometric or structural representations that make multivariate analogues tractable.
• Investigation of implications for adversarial robustness, empirical process theory, and optimal design of concentration inequalities.
• Deeper study of the connections between Gaussian rearrangement/isoperimetry and convex functional dominance.

Conclusion

The paper establishes that the optimal scaling constant $c$6 for convex order domination of one-dimensional MGF-sub-Gaussian random variables by a normal variable is $c$7. This result provides a definitive answer to an open question on the interplay between sub-Gaussianity and convex order, reveals a strict gap with the tail sense analogue, and lays groundwork for further investigations into sharp probabilistic comparison theorems in higher dimensions and general settings.