The sharp one-dimensional convex sub-Gaussian comparison constant

Abstract: Let $X$ be an integrable real random variable with mean zero and two-sided sub-Gaussian tail $\mathbb{P}(|X|>t)\le 2e^{-t^{2}/2}$ for all $t\ge 0$. We determine the smallest constant $c_\star$ such that $X$ is dominated in convex order by $c_\star G$, where $G$ is standard normal. Equivalently, $c_\star^2$ is the sharp one-dimensional convex sub-Gaussian comparison constant appearing in the \emph{Optimization Constants in Mathematics} repository~\cite{optimization-constants-repo}. We show that $c_\star$ is given by an explicit system of one-dimensional equations and is attained by an extremal distribution that saturates the tail constraint. Numerically, $c_\star \approx 2.30952$ (so $c_\star^2 \approx 5.33386$). We also determine the analogous sharp constant under a two-sided sub-exponential tail bound, with convex domination by a scaled Laplace law. Finally, we record two higher-dimensional consequences: a sequential tensorization principle for multivariate convex domination, and a dimension-free Gaussian comparator for the cone generated by convex ridge functions (the linear convex order).

• The paper determines the sharp convex sub-Gaussian constant (c*≈2.30952) that ensures any mean-zero sub-Gaussian variable is convexly dominated by a scaled Gaussian.
• It employs a stop-loss envelope construction and explicit one-dimensional integral equations to identify unique extremizers that saturate the tail constraints.
• The findings extend to higher-dimensional settings, providing precise tools for concentration inequalities, convex risk measures, and stochastic ordering.

The Sharp One-Dimensional Convex Sub-Gaussian Comparison Constant

Introduction and Context

This work addresses a precise sub-Gaussian comparison problem under convex ordering. Given an integrable, mean-zero real random variable $X$ with a two-sided sub-Gaussian tail bound, the authors determine the smallest constant $c_\star$ for which the convex order $X \preceq_{cx} c_\star G$ holds, $G$ denoting the standard normal law. This sharp comparison constant not only identifies the extremizers but also determines explicit numerical values with direct implications for stochastic order theory and high-dimensional probability.

The results refine a recent theorem by van Handel, which provides a universal comparison for multivariate sub-Gaussian random vectors (Handel, 21 Dec 2025). The sharp value of the comparison constant in dimension one has been unknown. This paper settles the question and identifies extremizers, with ramifications in the theory of convex orderings and probabilistic inequalities.

Characterization of the Comparison Constant

Under the two-sided tail constraint

$$\mathbb{P}(|X| > t) \le 2 e^{-t^2/2}, \qquad \forall t \ge 0,$$

and $\mathbb{E}[X] = 0$, the primary contribution is to find the minimal $c_\star$ such that

$X \preceq_{cx} c_\star G,$

that is, for all convex $f$ for which expectations are finite,

$\mathbb{E}[f(X)] \le \mathbb{E}[f(c_\star G)].$

The authors derive an explicit system of one-dimensional integral equations characterizing $c_\star$0. They show the following:

• The extremal distribution is unique and saturates the tail constraint everywhere.
• The calculation of $c_\star$1 proceeds via a envelope-maximization argument using stop-loss transforms, which characterize convex order in dimension one.
• Numerically, $c_\star$2; consequently, $c_\star$3.

A companion result calculates the sharp constant for two-sided sub-exponential tails with Laplace comparison, yielding $c_\star$4.

Methodology: Stop-Loss Envelope Construction

A key innovation is the explicit construction of a stop-loss envelope functional $c_\star$5 dominating the stop-loss transform of any mean-zero random variable constrained by a given symmetric tail $c_\star$6, recovering

$c_\star$7

with constants $c_\star$8 defined in terms of $c_\star$9. This construction enables tight, explicit upper bounds for

$X \preceq_{cx} c_\star G$0

Through the stop-loss representation of convex order and explicit calculation for the Gaussian comparator, the authors determine the precise $X \preceq_{cx} c_\star G$1 for which the dominated envelope property holds with equality.

The extremal law achieving equality is constructed: it is a three-part mixture with an absolutely continuous part up to a threshold and point masses at symmetric points, matching the tail at every $X \preceq_{cx} c_\star G$2.

Main Theorem

The main theorem is formalized as follows:

Let $X \preceq_{cx} c_\star G$3 be an integrable, mean-zero real random variable with $X \preceq_{cx} c_\star G$4 for all $X \preceq_{cx} c_\star G$5. Then for all convex $X \preceq_{cx} c_\star G$6,

$X \preceq_{cx} c_\star G$7

where $X \preceq_{cx} c_\star G$8 is specified via an explicit equation involving Gaussian integrals. For every $X \preceq_{cx} c_\star G$9, this bound is sharp: there exists $G$0 and convex $G$1 such that the inequality is reversed.

Sub-Exponential Analogue

A parallel argument is developed for variables with two-sided sub-exponential tails, i.e.,

$G$2

and convex domination by a rescaled Laplace law $G$3. The minimal constant is again characterized by an explicit integral system and computed as $G$4.

Higher-Dimensional Implications

Although the primary theorem is inherently one-dimensional, the authors document two higher-dimensional consequences:

1. Sequential Tensorization: If a random vector $G$5 admits a coordinate/martingale decomposition with sub-Gaussian marginal increment bounds, then the overall law is convex dominated by an appropriate product Gaussian law, with the sharp 1D constant $G$6. This is formalized via a sequential tensorization principle and conditional versions of the main result.
2. Cone Convex Order: For a mean-zero random vector $G$7 with $G$8 uniformly over all unit vectors $G$9, the law of $$\mathbb{P}(|X| > t) \le 2 e^{-t^2/2}, \qquad \forall t \ge 0,$$0 is convex dominated by $$\mathbb{P}(|X| > t) \le 2 e^{-t^2/2}, \qquad \forall t \ge 0,$$1 for expectations against all functions in the cone generated by nonnegative sums of univariate convex functions along directions (ridge-convex functions).

Both extensions rely critically on the one-dimensional nature of the optimal bound and its tensorization through convex order representations and martingale/conditional law decompositions.

Numerical Values and Sharpness

The authors compute:

• $$\mathbb{P}(|X| > t) \le 2 e^{-t^2/2}, \qquad \forall t \ge 0,$$2 (Gaussian case).
• $$\mathbb{P}(|X| > t) \le 2 e^{-t^2/2}, \qquad \forall t \ge 0,$$3 (Laplace case).

No part of the argument relies on nonconstructive existence; all constants and extremizers are given explicitly.

The bound is sharp; for every $$\mathbb{P}(|X| > t) \le 2 e^{-t^2/2}, \qquad \forall t \ge 0,$$4 one can construct $$\mathbb{P}(|X| > t) \le 2 e^{-t^2/2}, \qquad \forall t \ge 0,$$5 and convex $$\mathbb{P}(|X| > t) \le 2 e^{-t^2/2}, \qquad \forall t \ge 0,$$6 violating the convex order.

Theoretical and Practical Implications

This work closes a fundamental question in convex comparison under sub-Gaussian tail constraints and provides tools likely to be applicable in:

• Concentration inequalities for functions of dependent variables,
• Convex risk measures in statistics and finance,
• Dimension-free comparison theorems in probability,
• Sharp constants for stochastic process inequalities (as tabulated in [Optimization Constants in Mathematics repository (2026)]).

Open directions include:

• Higher-dimensional extension for genuine convex order (beyond ridge-convex cones).
• Exploration of sharper bounds under moment-generating function sub-Gaussianity (as the current work demonstrates via tail bounds).
• Extension to infinite-dimensional or Banach space–valued settings and other stochastic orders.

Conclusion

The authors’ work delivers the sharp one-dimensional convex sub-Gaussian comparison constant, characterizing both its value and the extremal distributions achieving it. The approach via stop-loss envelopes and explicit integral characterization leads to both theoretical insights and practical comparison inequalities with sharp constants. Extensions to higher-dimensional settings and other tail constraints are provided, with clear paths for further generalization in the structure of convex orderings and their extremal laws.