Chernoff's Density Is Strongly Log-Concave

Xianyang Zhang, Quan Zhou

arXiv:2607.18619v1 Primary: math.PR Secondary: math.ST

Abstract

Let $f$ be the density of the Chernoff random variable $\mathrm{argmax}_{t\in\mathbb{R}}\{W(t)-t^2\}$, where $W$ is a two-sided Brownian motion. This note proves the conjecture of Balabdaoui and Wellner (2014) that $f$ is strongly log-concave. The proof was generated in its entirety by GPT-5.6 Sol.

Disclosure

“ariable $\mathrm{argmax}_{t\in\mathbb{R}}\{W(t)-t^2\}$, where $W$ is a two-sided Brownian motion. This note proves the conjecture of Balabdaoui and Wellner (2014) that $f$ is strongly log-concave. The proof was generated in its entirety by GPT-5.6 Sol.”
arXiv metadata: abstract

Classification: Substantial proof generation
Multiplier: 10
Verified: 2026-08-23T18:59:43.589205Z

Structural counts

Pages 9 pdf

Theorems 1 source

Lemmas 2 source

Propositions 1 source

Corollaries 1 source

Definitions 0 source

Displayed equations 62 source

Bibliography entries 12 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file chernoff_strong_logconcavity_revised.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.