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Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

The paper establishes that any random variable with a moment generating function bounded by that of a standard normal distribution is dominated in convex order by the scaled normal variable $G/\mathbb{E}[|G|]$ , with equality achieved for the symmetric Bernoulli distribution and the absolute value function.

Original authors: Yihan Zhang

Published 2026-04-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yihan Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to guess the "worst-case scenario" for a random event. In the world of probability, we often deal with variables that behave nicely, like a bell curve (the Standard Gaussian). These variables are predictable: extreme values are rare, and most outcomes cluster around the middle.

But what if you have a different random variable, one that isn't a perfect bell curve but still behaves "nicely" in a specific way? Specifically, what if this variable is 1-sub-Gaussian? This is a fancy way of saying its "tail" (the chance of extreme values) is thin enough that it doesn't explode faster than a bell curve does.

The paper by Yihan Zhang asks a simple but deep question: If we have any random variable that behaves this nicely, how much do we need to "stretch" a standard bell curve to guarantee that it is always "bigger" or "more spread out" than our variable?

Here is the breakdown of the paper's journey, using everyday analogies.

1. The "Stretching" Game (Convex Order)

Think of two friends, X (your mysterious random variable) and G (a standard bell curve).
The paper is interested in a concept called Convex Order. Imagine you are a risk-averse investor. You have a "pain function" (a convex function) that measures how much you dislike volatility.

If X is "dominated" by G, it means that no matter how you measure the risk (how you twist your pain function), G will always hurt you more or equal to X.
In other words, G is the "worse" (more volatile) outcome.

The question is: If X is a 1-sub-Gaussian variable, how much do we need to stretch G (multiply it by a constant $c$ ) so that the stretched version ( $c \times G$ ) is guaranteed to be the "worse" outcome compared to X?

2. The Two Definitions of "Nice"

The paper notes that mathematicians have two slightly different ways to define a "nice" variable:

The Tail Definition: Looking at the probability of extreme events (like a storm hitting).
The Moment Generating Function (MGF) Definition: Looking at the "average energy" of the variable.

Previous research had solved the "stretching" problem for the Tail Definition, finding a specific number (about 2.31). But the author wondered: What is the number for the MGF definition? Is it the same? Is it bigger? Is it smaller?

3. The Big Discovery: The Magic Number

The paper proves that for the MGF definition, the magic stretching constant is exactly $\sqrt{\pi/2}$ (approximately 1.25).

This is a surprisingly small number! It means that if your variable behaves nicely according to the MGF rules, you only need to stretch the standard bell curve by about 25% to guarantee it covers all the risks of your variable. This is much tighter than the 2.31 factor found for the other definition.

Why is this number special?
The paper reveals that this number is actually the inverse of the "average absolute size" of a standard bell curve.

Imagine a standard bell curve. If you take the average of how far its points are from zero (ignoring direction), you get a number.
The magic constant is simply 1 divided by that average.
It turns out the "worst-case" variable that pushes this limit is a simple coin flip: a variable that is either -1 or +1 with equal probability. If you try to stretch the bell curve any less than 1.25 times, this coin flip will "break" the rule.

4. How They Solved It: The "Hinge" Trick

How did the author prove this? They used a clever mathematical shortcut.

The Hinge: Any "curved" shape (convex function) can be built out of simple "hinges" (like a door hinge that only opens one way).
The Reduction: Instead of checking every possible shape and every possible random variable, the author showed that you only need to look at the simplest possible variables: two-point distributions (like that coin flip).
The Geometry: They then compared the coin flip to the bell curve. To do this, they had to prove a new, very specific inequality about the shape of the bell curve (related to its "isoperimetric function," which is a fancy way of describing how the curve's edge behaves). They showed that the bell curve is "thick" enough to always cover the coin flip if stretched by $\sqrt{\pi/2}$ .

5. The Limitation: Why Not 3D?

The paper ends with a humble admission. This solution works beautifully in one dimension (a single line).
However, the mathematical tools used (the "hinge" trick) rely on the fact that we are dealing with a single line. In higher dimensions (like a 3D space), we don't have a simple way to break down complex shapes into hinges.
So, while we know the answer for a single number, we don't yet know the answer for a cloud of points in 3D space. The author leaves that as a mystery for future mathematicians.

Summary

The Problem: How much do we need to stretch a standard bell curve to safely cover any "nice" random variable?
The Answer: For the specific definition used here, the answer is $\sqrt{\pi/2}$ (approx 1.25).
The Method: The author simplified the problem by realizing the "worst-case" scenario is just a simple coin flip, and then proved the bell curve is big enough to cover it.
The Catch: This trick only works for single numbers, not for complex, multi-dimensional data.

1. Problem Statement

The paper addresses the problem of determining the optimal constant $c$ such that any centered, 1-sub-Gaussian random variable $X$ is dominated by a scaled standard Gaussian $cG$ in the convex order.

Definitions:
- 1-Sub-Gaussian (MGF sense): A random variable $X$ is 1-sub-Gaussian if its moment generating function (MGF) satisfies $M_X(\lambda) = \mathbb{E}[e^{\lambda X}] \leq e^{\lambda^2/2}$ for all $\lambda \in \mathbb{R}$ .
- Convex Order ( $X \preceq Y$ ): $X$ is dominated by $Y$ if $\mathbb{E}[f(X)] \leq \mathbb{E}[f(Y)]$ for all convex functions $f: \mathbb{R} \to \mathbb{R}$ (assuming expectations exist).
- Goal: Find the minimal constant $c_{\text{mgf}}$ such that for all $X$ satisfying the MGF condition, $X \preceq c_{\text{mgf}} G$ , where $G \sim \mathcal{N}(0, 1)$ .
Context: Previous work (e.g., Han [2025], Davis & Power [2026]) established the existence of such a constant but focused on "tail-based" sub-Gaussian definitions or higher dimensions. The optimal constant for the tail definition ( $c_{\text{tail}} \approx 2.31$ ) was known, but the constant for the MGF definition ( $c_{\text{mgf}}$ ) remained an open question, particularly regarding whether it differs from the tail-based constant.

2. Methodology

The proof relies on a combination of probabilistic reduction techniques and sharp analytical inequalities.

A. Reduction to Hinge Functions and Two-Point Distributions

Hinge Representation: The authors utilize the fact that any convex function $f$ can be represented as a linear combination of hinge functions $(x-t)_+ = \max(x-t, 0)$ . Consequently, proving $X \preceq Y$ reduces to showing $\mathbb{E}[(X-t)_+] \leq \mathbb{E}[(Y-t)_+]$ for all $t \in \mathbb{R}$ (Proposition 3).
Two-Point Reduction (Lemma 5): A crucial step is showing that for any sub-Gaussian $X$ $X$ and any threshold $t$ $t$ , the quantity $S_X(t) = \mathbb{E}[(X-t)_+]$ $S_{X} (t) = E [(X - t)_{+}]$ is maximized by a specific two-point distribution $Y$ $Y$ .
- $Y$ takes values $\mu_+$ and $\mu_-$ with probabilities $p$ and $1-p$ , where $p = \mathbb{P}(X > t)$ .
- The authors prove that if $X$ is 1-sub-Gaussian, this constructed $Y$ is also 1-sub-Gaussian and preserves the value of $S_X(t)$ .
- This reduces the infinite-dimensional optimization problem to analyzing a family of two-point distributions.

B. Optimization of the Two-Point Distribution

For a two-point variable $Y$ with mean 0 and MGF bounded by $e^{\lambda^2/2}$ , the authors derive an upper bound on the parameter $\nu$ (which scales the distance between the two points) in terms of the probability $p$ .

They establish the inequality:
$p(1-p)\nu \leq \sqrt{\frac{\pi}{2}} I(p)$
where $I(p) = \phi(\Phi^{-1}(p))$ is the Gaussian isoperimetric function (the density of the standard normal at the quantile corresponding to $p$ ).

C. Sharp Lower Bound on the Isoperimetric Function

To complete the proof, the authors need to show that the scaled Gaussian $cG$ satisfies the convex dominance condition. This requires a lower bound on the "stop-loss" function of the Gaussian, $S_{cG}(t)$ .

Key Technical Lemma (Lemma 2): The paper proves a sharp lower bound for the Gaussian isoperimetric function $I(p)$ :
$I(p) \geq \frac{2}{\sqrt{\pi}} p(1-p) \sqrt{\frac{\log(\frac{1-p}{p})}{1-2p}}$
This inequality is proven using a sophisticated analytical approach involving hyperbolic functions, Taylor expansions, and monotonicity arguments on auxiliary functions (Lemmas 6–8 in the Appendix). The constant $2/\sqrt{\pi}$ is shown to be sharp.

3. Key Results

Main Theorem (Theorem 1)

The optimal constant for the MGF-based sub-Gaussian definition is:
$c_{\text{mgf}} = \sqrt{\frac{\pi}{2}} \approx 1.25331$
This implies that for any 1-sub-Gaussian random variable $X$ (in the MGF sense), $X \preceq \sqrt{\frac{\pi}{2}} G$ .

Sharpness

The bound is tight. Equality is attained when:

$X \sim \text{Unif}(\{-1, 1\})$ (Rademacher distribution).
$f(x) = |x|$ .
In this case, $\mathbb{E}[|X|] = 1$ , while $\mathbb{E}[|cG|] = c \sqrt{2/\pi}$ . Setting $1 = c \sqrt{2/\pi}$ yields $c = \sqrt{\pi/2}$ .

Comparison with Tail-Based Constants

The paper highlights a significant distinction between definitions:

Tail-based ( $c_{\text{tail}}$ ): $\approx 2.30952$ (from Davis & Power [2026]).
MGF-based ( $c_{\text{mgf}}$ ): $\approx 1.25331$ .
The MGF definition yields a strictly smaller (and thus stronger) dominance constant. The paper notes that $c_{\text{mgf}} = 1/\mathbb{E}[|G|]$ .

4. Significance and Limitations

Interpretability: The result provides a clean, interpretable constant ( $1/\mathbb{E}[|G|]$ ) for the MGF definition, contrasting with the more complex constant derived for the tail definition.
Resolution of Open Question: It answers a question posed by Sam Power regarding the optimal constant for MGF-based sub-Gaussianity.
Limitation to Dimension 1: The proof heavily relies on the hinge representation of univariate convex functions (Equation 3). The authors explicitly state that a natural analogue of this representation does not exist in higher dimensions ( $d \geq 2$ ). Therefore, the method does not generalize to the multivariate case, leaving the sharp constant for $d \geq 2$ as an open problem.
Analytical Contribution: The derived lower bound on the Gaussian isoperimetric function (Lemma 2) is presented as a result of independent interest, offering a simple elementary function that tightly bounds $I(p)$ .

Conclusion

Yihan Zhang's paper establishes that the sharpest possible convex order domination for 1-sub-Gaussian random variables (defined via MGF) is achieved by scaling the standard Gaussian by $\sqrt{\pi/2}$ . The proof elegantly bridges probabilistic reduction (to two-point distributions) and deep analytical inequalities (on the Gaussian isoperimetric function), providing a definitive answer for the one-dimensional case while highlighting the challenges in extending these results to higher dimensions.