Probability · Statistics · Conjecture

Strong log-concavity of the Chernoff density

Does the Chernoff density have a uniformly positive lower bound on its negative log-curvature?

ProvedOpenMath assessment · Oct 14, 2026

Literature: 1 published paper · 1 preprint
Formal: None linked
Computation: None recorded
Review: Not yet reviewed

Overview

Known

The selected preprint proves existence of a uniformly positive curvature bound, beyond published ordinary log-concavity. It does not establish the sharper 2014 origin-minimum conjecture. Independent OpenMath mathematical review is pending.

Uniform positive curvature of the Chernoff densityProvedZhang & Zhou 2026

Latest

An analytic strong-curvature proof is submittedJul 21, 2026

Fix the Brownian scaling, define Z as a location, and distinguish ordinary log-concavity from a uniform positive curvature bound.

Precise statement · Exact conventions and hypotheses

Let W be two-sided standard Brownian motion with W(0) = 0, and $Z=\operatorname*{argmax}_{t\in\mathbb R}\{W(t)-t^2\}$ . For its positive smooth density f, write $\kappa(x)=-(\log f)^{\prime\prime}(x)$ . Strong log-concavity means there exists a constant c > 0 such that κ(x) ≥ c for every real x. This fixes the standard Brownian variance and unit quadratic drift; changing those conventions rescales the density and curvature.

Chernoff’s density is log-concave →

Definitions

Two-sided standard Brownian motion: A continuous random path on both sides of zero, with W(0) = 0 and independent standard Brownian motions on the positive and negative half-lines. Subtracting t² penalizes locations far from zero.
Chernoff’s Density Is Strongly Log-Concave →
Strong log-concavity: The potential −log f has second derivative at least a single positive constant everywhere. This is stronger than convexity, and stronger than merely positive curvature at every point.
Chernoff’s density is log-concave →
Chernoff’s Density Is Strongly Log-Concave →

Background

The earlier ordinary log-concavity theoremProved
Airy factorization of the density and curvatureProved

Question paraphrased by OpenMath. Original wording is available in the sources.

Landscape

Figures and explorable examples. Selecting a statement opens its exact wording.

The Chernoff density

The density of the location Z where W(t) − t² is largest: symmetric, with its mode at 0 and very light tails.

A numerical computation from the cited Airy formula, not a proof; plotted on [−2, 2].

The Airy factorisation used to compute f.
f is log-concave (2014).

Dataset chernoff-density-curve v1 · g and its first two derivatives by numerical inversion of the bilateral Laplace transform M(s) = Ai(0)/Ai(2^(−1/3)s) along the imaginary axis (mpmath quad, 20 digits) at x = −3, −2.95, …, 3. Checks: Simpson mass 1.000000000, symmetry, and κ(x) > 4|x| at every grid point; variance 0.2635596. Software: python 3.12.15, mpmath 1.3.0. 20-digit quadrature, stored as floating point; a numerical illustration. SHA-256 86d132590c72… · Download JSON

Data

Series	Points	Last value (x = 2)
f(x)	81	0.00012120178870205772

How curved is log f?

κ(x) = −(log f)″(x) stays above the proved lower bound 4|x| and is about 3.405 at the origin, so it is bounded away from zero: the strong log-concavity theorem.

Numerical values on [−2, 2] illustrate the theorems; they prove neither. Whether κ is smallest at 0 everywhere is the open origin-minimum question.

inf κ > 0: the theorem.
The proved bound κ(x) > 4|x|.
Open: is κ smallest at 0?

Dataset chernoff-log-curvature v1 · g and its first two derivatives by numerical inversion of the bilateral Laplace transform M(s) = Ai(0)/Ai(2^(−1/3)s) along the imaginary axis (mpmath quad, 20 digits) at x = −3, −2.95, …, 3. Checks: Simpson mass 1.000000000, symmetry, and κ(x) > 4|x| at every grid point; variance 0.2635596. Software: python 3.12.15, mpmath 1.3.0. 20-digit quadrature, stored as floating point; a numerical illustration. SHA-256 b2b5d1c7aa53… · Download JSON

Data

Series	Points	Last value (x = 2)
κ(x)	81	8.265109346118434
4\|x\| (proved bound)	81	8

Visual setup

A random path selects a location

An illustrative path with quadratic drift: Z is the location of the maximum, not the maximum height.

Reading boundary. This authored sketch is not a Brownian sample, a numerical Chernoff density or a verification of log-curvature.

Z: where W(t) − t² is largestQuestion scope

Z is the location of the maximum, not its height; the density of Z is the subject of the theorem.

Text version

The horizontal coordinate is t. A stylized continuous path represents W(t) − t², passes through zero at t = 0, and has its displayed maximum at Z. Repeating the random experiment produces the distribution whose density is f.

Layout is schematic; coordinates and line lengths carry no mathematical scale.

Z: where W(t) − t² is largest
Proved
Uniform positive curvature of the Chernoff density
Scope & assumptions. The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound.
Uniform positive curvature of the Chernoff density →

Sources & review

Authored by OpenMath source curation on 2026-10-14 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.

Chernoff’s density is log-concaveLog-concavity theorem and Conjecture 3.1PublishedOriginal passage ↗
Chernoff’s Density Is Strongly Log-ConcaveTheorem 1 and Section 5PreprintOriginal passage ↗

Results

The question 1

Proved
Uniform positive curvature of the Chernoff density
The lower bound is uniform over the whole real line.
Zhang & Zhou 2026
Conjecture · Primary formulation
ProvedZhang & Zhou 2026
Uniform positive curvature of the Chernoff density
For $Z=\operatorname*{argmax}_{t\in\mathbb R}\{W(t)-t^2\}$ , with W two-sided standard Brownian motion and density f, $\inf_{x\in\mathbb R}-(\log f)^{\prime\prime}(x)>0$ .
Assumptions
- The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound.
Distribution
Z = argmax{W(t)−t²}, standard W
Curvature requirement
κ(x) ≥ c for all x, for some c > 0
Uniform positive constant
Required
In this problem
The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound.
Proof outline
Pointwise strictness plus tail control
Exponential peeling, Airy convolution and compactness
The one-sided curvature v is strictly positive and satisfies v(x) ≥ 4x. Factorization gives κ(x) = v(x)+v(−x) > 4|x|. Continuity controls the remaining compact interval, yielding a uniform positive lower bound.
The proof does not report an optimal numerical constant. Authors’ checks and a manuscript argument are separate from proof-assistant execution.
Proved: Strong log-concavity proved in the selected preprint ↗ · source dated Jul 21, 2026 · observed Oct 14, 2026. Standard two-sided Brownian motion with unit quadratic drift; no explicit optimal uniform constant is asserted.
Depends on: Airy factorization of the density and curvature · Pointwise and tail curvature bounds
- Chernoff’s Density Is Strongly Log-Concave →
Independent mathematical assessment of an argument remains separate from the statement’s attributed status.
Sources
- Chernoff’s density is log-concave →
- Chernoff’s Density Is Strongly Log-Concave →
Open statement page →

Bounds 1

Proved
Pointwise and tail curvature bounds
Continuity, strict positivity on compact intervals and growth in the tails together yield the uniform lower bound.
> 4|x|Zhang & Zhou 2026
Theorem · Bound
ProvedZhang & Zhou 2026
Pointwise and tail curvature bounds
For the standard Chernoff density f, κ(x) = −(log f)″(x) satisfies $\kappa(x)>4|x|$ for every real x, and $\inf_{x\in\mathbb R}\kappa(x)>0$ .
Assumptions
- Continuity, strict positivity on compact intervals and growth in the tails together yield the uniform lower bound. The numerical optimum of inf κ is not given.
In this problem
Continuity, strict positivity on compact intervals and growth in the tails together yield the uniform lower bound. The numerical optimum of inf κ is not given.
Bound
Negative log-curvature κ(x)
> 4|x|
Conditions
Standard Brownian variance, drift t² and every real x; the inequality is strict.
Source date
Jul 21, 2026
- Chernoff’s Density Is Strongly Log-Concave →
Sources
- Chernoff’s Density Is Strongly Log-Concave →
Open statement page →

Variants and analogues 1

Open
The sharper origin-minimum curvature conjecture
This is stronger than inf κ > 0. The selected 2026 theorem does not prove that the origin minimizes curvature, and this curation does not assert a comprehensive current-status assessment.
Conjecture · Different statement · related variant
Open
The sharper origin-minimum curvature conjecture
For the standard Chernoff density f, $\kappa(x)=-(\log f)^{\prime\prime}(x)$ satisfies $\kappa(x)\ge\kappa(0)$ for every real x.
Assumptions
- Two-sided standard Brownian motion and quadratic drift t².
- The proposed optimal constant is the curvature at the origin; κ(0) ≈ 3.4052 is a numerical value reported in the 2014 source.
Distribution
Z = argmax{W(t)−t²}, standard W
Curvature requirement
κ(x) ≥ κ(0) for every x
Uniform positive constant
Specified by κ(0); numerical value ≈ 3.4052
In this problem
This is stronger than inf κ > 0. The selected 2026 theorem does not prove that the origin minimizes curvature, and this curation does not assert a comprehensive current-status assessment.
Sources
- Chernoff’s density is log-concave →
Open statement page →

Background 2

Proved
The earlier ordinary log-concavity theorem
Ordinary log-concavity requires nonnegative curvature, without a uniform strictly positive lower bound.
Balabdaoui & Wellner 2014
Theorem · Supporting result
ProvedBalabdaoui & Wellner 2014
The earlier ordinary log-concavity theorem
The density of argmaxₜ∈ℝ{W(t)−t²}, for two-sided standard Brownian motion W, is log-concave.
Assumptions
- Ordinary log-concavity requires nonnegative curvature, without a uniform strictly positive lower bound.
Distribution
Z = argmax{W(t)−t²}, standard W
Curvature requirement
κ(x) ≥ 0 for all x
Uniform positive constant
Not required
In this problem
Ordinary log-concavity requires nonnegative curvature, without a uniform strictly positive lower bound.
Sources
- Chernoff’s density is log-concave →
Open statement page →
Proved
Airy factorization of the density and curvature
The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero.
Zhang & Zhou 2026
Theorem · Supporting result
ProvedZhang & Zhou 2026
Airy factorization of the density and curvature
For a positive normalized factor g with bilateral Laplace transform M(s) = Ai(0)/Ai(2^(−1/3)s), the standard Chernoff density has f(x) = C_f g(x)g(−x), where C_f = 2^(−1/3)/Ai(0)², and κ(x) = v(x)+v(−x) for v = −(log g)″.
Assumptions
- The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives.
In this problem
The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives.
Sources
- Chernoff’s Density Is Strongly Log-Concave →
Open statement page →

Compare statements · 3

Each row is a separate assertion. Its assumptions determine which question it answers.

Statement scopes and hypotheses
Statement	Distribution	Curvature requirement	Uniform positive constant
The earlier ordinary log-concavity theorem → Proved Exact statement & sources	Z = argmax{W(t)−t²}, standard W	κ(x) ≥ 0 for all x	Not required
Uniform positive curvature of the Chernoff density → Proved Exact statement & sources	Z = argmax{W(t)−t²}, standard W	κ(x) ≥ c for all x, for some c > 0	Required
The sharper origin-minimum curvature conjecture → Open Exact statement & sources	Z = argmax{W(t)−t²}, standard W	κ(x) ≥ κ(0) for every x	Specified by κ(0); numerical value ≈ 3.4052

The earlier ordinary log-concavity theorem →

Chernoff’s density is log-concave →

Uniform positive curvature of the Chernoff density →

The sharper origin-minimum curvature conjecture →

Chernoff’s density is log-concave →

Connections

How the recorded statements connect

━ ⇔ equivalent
→ implies
┅ analogue or related (no logical link)
Border colour = status

2 connections · 1 consequence. Select a statement in the graph to show only its entries.

Analogues and related questions 2

Ordinary log-concavity related to Uniform positive curvature.
Scope comparison
Ordinary and strong log-concavity. The newer theorem strengthens ordinary log-concavity for the same distribution to a uniform strictly positive curvature bound.
Limit A strictly positive pointwise second derivative alone does not give a uniform lower bound.
Sources & review
- Chernoff’s density is log-concave ↗Log-concavity theorem and Conjecture 3.1
- Chernoff’s Density Is Strongly Log-Concave ↗Theorem 1 and Section 5
Independent relationship review: not recorded. Link to this connection
Uniform positive curvature related to Minimum curvature at the origin.
Variant
Uniform positivity and the sharp constant. The qualitative question asks for some uniform positive constant; the sharper 2014 conjecture specifies the origin curvature as the best constant.
Limit The selected strong-log-concavity proof gives no origin-minimum conclusion. The old conjecture alone is not a current literature assessment.
Sources & review
- Chernoff’s density is log-concave ↗Introduction; Section 3, Conjecture 3.1
- Chernoff’s Density Is Strongly Log-Concave ↗Theorem 1 and Section 5
Independent relationship review: not recorded. Link to this connection

Consequences 1

Assuming Uniform positive curvatureProved

Control over the whole distribution
Under stated assumptions
The stronger theorem establishes a single positive curvature lower bound across the real line. The source explains why this places Chernoff’s law in the class supporting Gaussian-type concentration and associated functional inequalities.
Limit The theorem does not say the law is Gaussian or supply an optimal concentration constant.
- Pointwise and tail curvature boundsProved
Sources & review
Supporting statements
- Uniform positive curvature of the Chernoff density →
- Pointwise and tail curvature bounds →
- Chernoff’s Density Is Strongly Log-Concave ↗Theorem 1 and Section 5
Source-grounded explanation · independent explanation review: pending.

Timeline

Jul 21, 2026An analytic strong-curvature proof is submittedRelated result
Zhang and Zhou submit the selected preprint, crediting GPT-5.6 Sol and their own checking and exposition work.
This is the arXiv submission date. The selected manuscript’s title page says 24 August 2026; neither is asserted to be the discovery date.
- Uniform positive curvature of the Chernoff densityProved
- Pointwise and tail curvature boundsProved
- Chernoff’s Density Is Strongly Log-Concave ↗Theorem 1 and Section 5
Also on the Frontier: Strong log-concavity of the Chernoff density
2014Log-concavity is proved and a stronger property proposedRelated result
Balabdaoui and Wellner publish ordinary log-concavity and conjecture uniform positive curvature.
This is a selected source-supported milestone, not an exhaustive history.
- The earlier ordinary log-concavity theoremProved
- Chernoff’s density is log-concave ↗Log-concavity theorem and Conjecture 3.1

Full history with sources →

Evidence

Evidence by kind
Literature	1 published paper · 1 preprint Status OpenMath assessment · Oct 14, 2026. Proved: Chernoff’s Density Is Strongly Log-Concave · Strong log-concavity proved in the selected preprint ↗ · source dated Jul 21, 2026 · checked Oct 14, 2026Standard two-sided Brownian motion with unit quadratic drift; no explicit optimal uniform constant is asserted.
Formal	None linked Reported in sources: No complete proof-assistant artifact was identified in the selected sources. The analytic manuscript and author checking are cited separately from independent mathematical assessment.
Computation	None recorded
Review	Not yet reviewed OpenMath assessment: The selected preprint proves existence of a uniformly positive curvature bound, beyond published ordinary log-concavity. It does not establish the sharper 2014 origin-minimum conjecture. Independent OpenMath mathematical review is pending. Source comparison and presentation; independent mathematical review remains pending. OpenMath pilot collection · OpenMath source curation · assessed Oct 14, 2026 · review due Jan 14, 2027 Collection last checked Oct 14, 2026.

Sources

Choose a starting point below. Each result links to its exact statement, cited passages and checks.

Start here

Original question
Chernoff’s density is log-concave →
Read the published theorem and the stronger conjecture.
Proof source
Chernoff’s Density Is Strongly Log-Concave →
Read Theorem 1 and the compactness argument.
Proof source
Chernoff’s Density Is Strongly Log-Concave →
Inspect the normalized factor, transform domain and prior Airy identity.

Bibliography 2

Chernoff’s density is log-concave →
Fadoua Balabdaoui · Jon A. Wellner · Published · Feb 5, 2014
Published Bernoulli 20(1), 231–244; ordinary log-concavity and the stronger curvature conjecture.
Chernoff’s Density Is Strongly Log-Concave →
Xianyang Zhang · Quan Zhou · Preprint · Jul 21, 2026
Theorem 1; reciprocal-Airy representation, exponential peeling, convolution identity, compactness and AI Usage.

Selected literature; source observations and coverage limits are retained in the publication details.

Contributors

Fadoua Balabdaoui and Jon A. Wellner proved ordinary log-concavity and posed the stronger conjecture. Xianyang Zhang and Quan Zhou authored the 2026 note; they credit GPT-5.6 Sol with the proof and report checking it, revising the exposition and recognizing prior work on an Airy identity. Chernoff and Groeneboom supplied foundational distribution and factorization results; Menon and Srinivasan supplied the earlier convolution identity used in the final proof.

Authors & roles

Chernoff’s density is log-concave →

Fadoua Balabdaoui · Jon A. Wellner

No AI contribution is reported in this selected source.

Chernoff’s Density Is Strongly Log-Concave →

Xianyang Zhang · Quan Zhou

The authors attribute the proof to GPT-5.6 Sol, describe an initial algebraic failure and later corrected approaches, and report carefully checking the argument and revising the exposition.

These are source credits; they do not imply OpenMath accounts, endorsements or independent mathematical review.

Record details

Record version: 8
Classification: MSC2020 60E05 / 62G20
Prerequisite level: graduate

Independent mathematical review is pending. External certificates and manuscript proofs are attributed to their sources.

Source observations & version history →

Overview

Known

Related open questions

Latest

Definitions

Background

Landscape

The Chernoff density

How curved is log f?

A random path selects a location

Uniform positive curvature of the Chernoff density

Results

The question 1

Assumptions

In this problem

Proof outline

Sources

Bounds 1

Assumptions

In this problem

Bound

Sources

Variants and analogues 1

Assumptions

In this problem

Sources

Background 2

Assumptions

In this problem

Sources

Assumptions

In this problem

Sources

The earlier ordinary log-concavity theorem →

Uniform positive curvature of the Chernoff density →

The sharper origin-minimum curvature conjecture →

Connections

Analogues and related questions 2

Consequences 1

Timeline

Evidence

Sources

Start here

Bibliography 2

Contributors

Chernoff’s density is log-concave →

Chernoff’s Density Is Strongly Log-Concave →