Chernoff's Density Is Strongly Log-Concave

Xianyang Zhang; Quan Zhou

Chernoff's Density Is Strongly Log-Concave

Probability 2026-07-21 v1 Statistics Theory

Authors: Xianyang Zhang , Quan Zhou

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.

Cite

@article{arxiv.2607.18619,
  title  = {Chernoff's Density Is Strongly Log-Concave},
  author = {Xianyang Zhang and Quan Zhou},
  journal= {arXiv preprint arXiv:2607.18619},
  year   = {2026}
}