Proved
Published ordinary log-concavity theorem ↗
The standard Chernoff density.
Chernoff’s density is log-concave · Log-concavity theorem and Conjecture 3.1 →
Source dated Feb 5, 2014 · Observed Oct 14, 2026
Theorem · version 1
The earlier ordinary log-concavity theorem
Statement
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
Evidence
Source reports
Connections
In this problem · publication v8 · Selected context
Strong log-concavity of the Chernoff density →
Supporting result · Proved
Related statements
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.
Uniform positive curvature of the Chernoff density →
Proved · version 1
For , with W two-sided standard Brownian motion and density f, .
- 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 →Relationship not independently reviewed. This concerns its use in this problem.
Sources
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.
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.
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.
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.
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
Published ordinary log-concavity theorem ↗
The standard Chernoff density.
Source dated Feb 5, 2014 · Observed Oct 14, 2026 · Source passage →
Record details
- Claim ID
- claim-chernoff-ordinary-log-concavity
- Statement version
- 1
- SHA-256
- 556eb81ea757177dff8c7300d47578bd08bde12306f5e0a6c8c8f51e163fad27