Anti-concentration inequalities for log-concave variables on the real line
Tulio Gaxiola, James Melbourne, Vincent Pigno, Emma Pollard
TL;DR
The paper addresses sharp anti-concentration for log-concave random variables on the real line, both in the Lebesgue and counting measures. It introduces a density-crossing and majorization approach centered on mode-centered asymmetric Laplace distributions to obtain universal variance-type bounds of the form $2\,\mathrm{Var}(X) \le \frac{1}{f(t)^2} + (\mu - t)^2$, with optimizations at the mode yielding $2\,\mathrm{Var}(X) \le \frac{1}{\|f\|_\infty^2} + (\mu - m)^2$, and equalities achieved by the corresponding Laplace laws. The discrete (on $\mathbb{Z}$) case derives analogous bounds, including $2\,\mathrm{Var}(Y) \le (\frac{1}{\mathbb{P}(Y=n)^2} - 1) + (\mu - n)^2$, with equality for discrete asymmetric Laplace and geometric distributions, and accompanying higher-moment bounds. A general density-crossing majorization framework underpins both continuous and discrete results, providing a streamlined, localization-free route to Karlin–Novikoff-type orderings and Orlicz-norm comparisons. The findings yield a cohesive, elementary toolkit for anti-concentration phenomena in log-concave settings, with clear equality cases and connections to known entropy-power and concentration bounds.
Abstract
We prove sharp anti-concentration results for log-concave random variables on the real line in both the discrete and continuous setting. Our approach is elementary and uses majorization techniques to recover and extend some recent and not so recent results.