Near-Gaussian counterexamples to the Ball-Nayar-Tkocz entropy concavity conjecture

Abstract

The Ball-Nayar-Tkocz conjecture asserts that, for independent real random variables $X,Y$ with a common log-concave density, the function $t\mapsto h(\sqrt tX+\sqrt{1-t}Y)$ is concave on $[0,1]$. Gaussian distributions have this property. We construct strongly log-concave counterexamples arbitrarily close to the Gaussian and study their persistence under Gaussian smoothing. Let $X$ have mean zero and variance one, let $G$ be an independent standard Gaussian, and let $q\in[0,1]$ denote the signal variance proportion in the smoothed variable $\sqrt qX+\sqrt{1-q}G$. In the asymmetric case, for every fixed $0<q\le1$, there are counterexamples for which the entropy of the weighted sum of two independent copies of the smoothed variable has strictly positive second derivative near the endpoints. In the symmetric case, for every $4/7<q\le1$, there are counterexamples for which an unequally weighted sum has strictly greater entropy than the equally weighted sum. Both families have smooth, strictly positive densities and are strongly log-concave before and after smoothing. The density ratios $f/\varphi$ converge uniformly to one, and each derivative of fixed positive order converges uniformly to zero, where $\varphi$ is the standard Gaussian density. The symmetric counterexamples can also match any prescribed finite number of Gaussian moments exactly. The asymmetric construction uses an endpoint expansion of entropy, whereas the symmetric construction exploits the different decay rates of high-order Hermite perturbations under different weights. Thus, densities violating concavity approach the Gaussian density in the strong sense described above. The question of uniform concavity in the symmetric class for $0<q\le4/7$ remains open.