Strong log-concavity of Chernoff’s density
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Chernoff’s distribution is the law of argmaxₜ{W(t)−t²} for two-sided Brownian motion W. Is its density strongly log-concave, meaning −(log f)″ has a strictly positive global lower bound?
Why it matters and what counts as progress
Why it matters: Strong log-concavity gives a uniform curvature property beyond the ordinary log-concavity already proved for this Brownian argmax law.
What counts as progress: Prove a uniform positive curvature bound, quantify it through the Airy factorization or strengthen tail-dependent curvature estimates.
How the problem developed
- 1 Feb 2014 · Conjecture [1]
Fadoua Balabdaoui and Jon A. Wellner Proved ordinary log-concavity of Chernoff’s density and conjectured that it is strongly log-concave.
- 21 Jul 2026 · Theorem claim [2]
Zhang and Zhou Reported a proof of strong log-concavity through curvature bounds for the Airy-function factor in Chernoff’s density.
AI contribution
Chernoff's Density Is Strongly Log-Concave
Theorem 1 proves that the one-sided Airy factor g has v(x) = -(log g)''(x) > 0 and v(x) ≥ 4x. Since Chernoff's density f is proportional to g(x)g(-x), its curvature κ(x) exceeds 4|x| and has a positive global infimum. This proves strong log-concavity of the density of argmax_t{W(t)-t²}.
Sources
Website timeline
Publication dates and editorial changes.
GPT-5.6 proof establishes strong log-concavity of Chernoff's density