Uniform positive curvature of the Chernoff density

Statement

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

Explanation

Proof outlines · 1

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.

Independent mathematical assessment of this argument remains separate from the statement’s attributed status.

Uniform positive curvature of the Chernoff density →

Chernoff’s Density Is Strongly Log-Concave →

Evidence

Source reports

Proved

Strong log-concavity proved in the selected preprint ↗

Standard two-sided Brownian motion with unit quadratic drift; no explicit optimal uniform constant is asserted.

Chernoff’s Density Is Strongly Log-Concave · Theorem 1 and Section 5 →

Source dated Jul 21, 2026 · Observed Oct 14, 2026

Connections

In this problem · publication v8

Strong log-concavity of the Chernoff density →

Primary formulation · Proved

Related statements

The earlier ordinary log-concavity theorem

The newer theorem strengthens ordinary log-concavity for the same distribution to a uniform strictly positive curvature bound.

A strictly positive pointwise second derivative alone does not give a uniform lower bound.

Details & sources

The earlier ordinary log-concavity theorem →

Proved · version 1

The density of argmaxₜ∈ℝ{W(t)−t²}, for two-sided standard Brownian motion W, is log-concave.

Ordinary log-concavity requires nonnegative curvature, without a uniform strictly positive lower bound.

Scope: Ordinary log-concavity requires nonnegative curvature, without a uniform strictly positive lower bound.

Chernoff’s density is log-concave →

Uniform positive curvature of the Chernoff density →

Proved · version 1

For $Z=\operatorname*{argmax}_{t\in\mathbb R}\{W(t)-t^2\}$ , with W two-sided standard Brownian motion and density f, $\inf_{x\in\mathbb R}-(\log f)^{\prime\prime}(x)>0$ .

The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound.

Scope: The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound.

Relationship not independently reviewed.

Connection in Strong log-concavity of the Chernoff density →

Related statements

This statement: Uniform positive curvature of the Chernoff density

The qualitative question asks for some uniform positive constant; the sharper 2014 conjecture specifies the origin curvature as the best constant.

The selected strong-log-concavity proof gives no origin-minimum conclusion. The old conjecture alone is not a current literature assessment.

Details & sources

Uniform positive curvature of the Chernoff density →

Proved · version 1

For $Z=\operatorname*{argmax}_{t\in\mathbb R}\{W(t)-t^2\}$ , with W two-sided standard Brownian motion and density f, $\inf_{x\in\mathbb R}-(\log f)^{\prime\prime}(x)>0$ .

The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound.

Scope: The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound.

The sharper origin-minimum curvature conjecture →

Open · version 1

For the standard Chernoff density f, $\kappa(x)=-(\log f)^{\prime\prime}(x)$ satisfies $\kappa(x)\ge\kappa(0)$ for every real x.

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.

Scope: 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.

Chernoff’s density is log-concave →

Relationship not independently reviewed.

Connection in Strong log-concavity of the Chernoff density →

Relationship not independently reviewed. This concerns its use in this problem.

Sources

Chernoff’s Density Is Strongly Log-Concave →

Theorem 1 and Section 5 → · Original passage ↗
Recorded wording
The paper establishes v(x) > 0 and v(x) ≥ 4x for the normalized one-sided factor, hence κ(x) > 4|x| and inf κ > 0 for the Chernoff density.
paraphrase · Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.
Equations (2)–(6); Sections 2–4 → · Original passage ↗
Recorded wording
The Chernoff density factors as f(x) = C_f g(x)g(−x), where the normalized one-sided factor has reciprocal-Airy bilateral Laplace transform. The curvature is v(x)+v(−x).
paraphrase · Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.

Xianyang Zhang · Quan Zhou · Preprint · Publication date Jul 21, 2026

Source-reported results and external checks are separate from independent OpenMath assessment.

Revision & attribution

Source revision: v1. Observed Oct 14, 2026.

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.

Chernoff’s density is log-concave →

Log-concavity theorem and Conjecture 3.1 → · Original passage ↗
Recorded wording
Ordinary log-concavity is established, and uniform positive negative log-curvature is proposed as a stronger property of the same Chernoff density.
paraphrase · Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.
Introduction; Section 3, Conjecture 3.1 → · Original passage ↗
Recorded wording
The sharper original conjecture requires κ(x) ≥ κ(0) for every x. Its reported numerical value κ(0) ≈ 3.4052 is not established as a uniform lower bound by the selected 2026 theorem.
paraphrase · Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.

Fadoua Balabdaoui · Jon A. Wellner · Published · Publication date Feb 5, 2014

Source-reported results and external checks are separate from independent OpenMath assessment.

Revision & attribution

Source revision: v2. Observed Oct 14, 2026.

No AI contribution is reported in this selected source.

History

Statement versions

Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.

Assessment history · 1

Assessments retained in these publication records. Earlier statement versions retain their own source histories.

Proved

Strong log-concavity proved in the selected preprint ↗

Standard two-sided Brownian motion with unit quadratic drift; no explicit optimal uniform constant is asserted.

Source dated Jul 21, 2026 · Observed Oct 14, 2026 · Source passage →

Record details

Claim ID: claim-chernoff-strong-log-concavity
Statement version: 1
SHA-256: c2299520080f575923e729e25686db80440ad6571e5b0fe46a4fcd5f15336ed2

Publication records

Strong log-concavity of the Chernoff density · publication v8 · JSON →