Chernoff's Density Is Strongly Log-Concave

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.