Chernoff's Density is Log-Concave

  Abstract: We show that the density of Z = argmax{W(t)−t 2}, sometimes known as Chernoff’s density, is log-concave. We conjecture that Chernoff’s density is strongly log-concave or “super-Gaussian”, and provide evidence in support of the conjecture. We also show that the standard normal density can be written in the same structural form as Chernoff’s density, make connections with L. Bondesson’s class of hyperbolically completely monotone densities, and identify a large sub-class thereof having log-transforms to R which are strongly log-concave. AMS 2000 subject classifications: Primary 60E05, 62E10; secondary 60E10, 60J65. Keywords and phrases: Airy function, Brownian motion, correlation inequalities, hyperbolically monotone, log-concave, monotone function estimation, Prekopa-Leindler theorem, Polya frequency function, Schoenberg’s theorem, slope process, strongly log-concave.