The sharp one-dimensional convex sub-Gaussian comparison constant
Pith reviewed 2026-05-13 18:17 UTC · model grok-4.3
The pith
Any mean-zero sub-Gaussian random variable is convex-dominated by a standard normal scaled by an explicit constant c_star ≈ 2.30952.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Any integrable mean-zero random variable X satisfying P(|X|>t) ≤ 2e^{-t²/2} for all t ≥ 0 is dominated in convex order by c_star G, where G is standard normal; the sharp c_star is the solution to an explicit one-dimensional system of equations attained by an extremal distribution that saturates the tail constraint.
What carries the argument
The extremal distribution that saturates the sub-Gaussian tail bound exactly at the points dictated by the convex-order optimality conditions.
If this is right
- The sharp constant equals approximately 2.30952, so its square is approximately 5.33386.
- An analogous sharp constant exists for two-sided sub-exponential tails, with domination by a scaled Laplace law.
- Sequential tensorization holds for multivariate convex domination under the same one-dimensional tail bounds.
- A dimension-free Gaussian comparator exists for the cone of convex ridge functions under linear convex order.
Where Pith is reading between the lines
- The one-dimensional reduction may extend to other tail families or stochastic orders by constructing analogous saturating distributions.
- The explicit numerical value supplies a concrete benchmark for testing concentration or moment-comparison inequalities in one dimension.
- The tensorization and ridge-function results suggest the constant can be lifted to certain classes of multivariate problems without dimension-dependent loss.
Load-bearing premise
An extremal distribution exists that simultaneously saturates the tail bound at the points required by the optimality conditions and solves the resulting one-dimensional system of equations.
What would settle it
A concrete integrable mean-zero random variable obeying the sub-Gaussian tail bound whose convex-order minimal scaling to a normal exceeds 2.30952, or an inconsistency in the proposed system of equations when the tail-saturation conditions are imposed.
read the original 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).
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper determines the smallest constant c_⋆ such that any integrable mean-zero random variable X with two-sided sub-Gaussian tails P(|X|>t) ≤ 2e^{-t²/2} for all t≥0 is dominated in convex order by c_⋆ G (G standard normal). Equivalently, c_⋆² is the sharp one-dimensional convex sub-Gaussian comparison constant. The authors show that c_⋆ solves an explicit system of one-dimensional equations and is attained by an extremal distribution saturating the tail bound at finitely many points, with numerical value c_⋆ ≈ 2.30952. Analogous results are obtained for sub-exponential tails (domination by scaled Laplace), together with two higher-dimensional consequences: a sequential tensorization principle and a dimension-free Gaussian comparator for the cone of convex ridge functions.
Significance. If the central claims hold, the result supplies the exact sharp constant for one-dimensional convex sub-Gaussian comparison, resolving an entry in the Optimization Constants repository. The reduction to a solvable one-dimensional system and the explicit extremal distribution are technically clean and immediately usable for computations. The higher-dimensional corollaries (tensorization and ridge-function comparison) extend the utility beyond the one-dimensional setting.
major comments (2)
- [Section 3] Section 3: the derivation imposes saturation P(|X|>t_i)=2e^{-t_i²/2} at finitely many points together with first-order optimality conditions from convex order. For the candidate extremal (typically discrete) measure to be admissible, the tail inequality must hold for every t, not merely at the t_i. The manuscript does not contain an explicit verification or argument that the solved distribution remains below the sub-Gaussian envelope on the complementary intervals; without this step the constructed object may lie outside the feasible set, so the reported c_⋆ is not guaranteed to dominate all admissible X.
- [Section 4] Section 4: the numerical value c_⋆ ≈ 2.30952 is stated without reported precision, solver method, or convergence diagnostics. Because the system is nonlinear and the constant is claimed to be sharp, the manuscript should supply the precise algebraic system, the numerical procedure, and an a-posteriori error bound confirming that the reported digits are accurate to at least four decimal places.
minor comments (2)
- [Section 1] The abstract and introduction use the phrase 'two-sided sub-Gaussian tail' without restating the precise inequality; a single displayed equation at the beginning of Section 1 would improve readability.
- [Section 5] In the discussion of the sub-exponential case, the scaling of the Laplace comparator is stated but the corresponding system of equations is only sketched; a compact display of the analogous one-dimensional system would make the parallel with the Gaussian case immediate.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will make the indicated revisions to strengthen the paper.
read point-by-point responses
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Referee: [Section 3] Section 3: the derivation imposes saturation P(|X|>t_i)=2e^{-t_i²/2} at finitely many points together with first-order optimality conditions from convex order. For the candidate extremal (typically discrete) measure to be admissible, the tail inequality must hold for every t, not merely at the t_i. The manuscript does not contain an explicit verification or argument that the solved distribution remains below the sub-Gaussian envelope on the complementary intervals; without this step the constructed object may lie outside the feasible set, so the reported c_⋆ is not guaranteed to dominate all admissible X.
Authors: We acknowledge that an explicit global verification of the tail bound is missing from the current presentation. While the first-order conditions derived from convex order, combined with the convexity of the sub-Gaussian envelope and the discrete support of the extremal measure, imply that the bound holds between saturation points, we agree this requires a dedicated argument. In the revised manuscript we will insert a short lemma in Section 3 that verifies the inequality on the complementary intervals by direct comparison of the cumulative distribution functions and the monotonicity properties of the relevant functions. This will rigorously confirm admissibility of the candidate measure. revision: yes
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Referee: [Section 4] Section 4: the numerical value c_⋆ ≈ 2.30952 is stated without reported precision, solver method, or convergence diagnostics. Because the system is nonlinear and the constant is claimed to be sharp, the manuscript should supply the precise algebraic system, the numerical procedure, and an a-posteriori error bound confirming that the reported digits are accurate to at least four decimal places.
Authors: We agree that the numerical section requires more detail to substantiate the claimed value. In the revision we will expand Section 4 to state the exact algebraic system of equations, describe the nonlinear solver (including tolerances and initialization), report convergence diagnostics such as residual norms, and supply an a-posteriori error bound (via interval arithmetic or residual analysis) that certifies accuracy to at least four decimal places. This will enable full reproducibility. revision: yes
Circularity Check
No circularity detected in the derivation chain
full rationale
The paper derives c_star by constructing an explicit finite system of one-dimensional equations from the first-order optimality conditions of convex order together with pointwise saturation of the sub-Gaussian tail bound at the support points of a candidate discrete extremal measure. This system is solved numerically to obtain the reported value. The construction does not reduce the claimed constant to a fitted parameter or to any self-cited prior result by definition; the equations are obtained directly from the problem statement without circular substitution. The cited repository merely records the constant and is not used to justify the derivation. The approach is therefore self-contained against the external definition of convex order and the tail constraint.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Definition of convex order: X ≼_cx Y iff E[φ(X)] ≤ E[φ(Y)] for every convex φ
- domain assumption Two-sided sub-Gaussian tail bound P(|X|>t) ≤ 2 exp(−t²/2) for all t ≥ 0
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
We show that c_⋆ is given by an explicit system of one-dimensional equations and is attained by an extremal distribution that saturates the tail constraint.
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
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[1]
[AS64] Milton Abramowitz and Irene A. Stegun, editors.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 ofApplied Mathematics Series. National Bureau of Standards, Washington, D.C., 1964. [DITc26] Damek Davis, Paata Ivanisvili, Terence Tao, and contributors. Optimization constants in mathematics. GitHub repositor...
work page 1964
discussion (0)
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