One-sided concentration near the mean of log-concave distributions
Iosif Pinelis
TL;DR
The paper addresses one-sided concentration near the mean for log-concave distributions with $E X=0$ and $E X^2=1$, deriving an explicit universal lower bound $P(0<X<\delta)\ge p(\delta)=\frac{\delta}{72(1+c\delta)}$ with $c=\frac{419}{100}$ and showing $f_X(0)\ge\frac{1}{72}$ as $\delta\to0$. The proof combines a median-based decomposition, yielding $2E|X|\ge1$, with a sharp pointwise bound $r_{\delta,b}(x)$, $r_{\delta,b}(x)=a(\delta,b)(\frac{x^2}{2}-b\frac{x^3}{3})$, where $a(\delta,b)=\min(\frac{16b}{3},6b^2\delta)$, and optimizes over $b$ while treating two monotonicity regimes of the pdf on $[0,\infty)$. The work situates the result relative to BKP bounds and discusses potential nonoptimality, supported by extremal considerations from the exponential case $Y\sim\mathrm{Exp}(1)$ with $Y-1\in L$, which attains the corresponding extremal values. Overall, the paper provides a concrete, albeit possibly improvable, bound on near-mean concentration for log-concave distributions and highlights avenues for tightening these inequalities.
Abstract
A lower bound on the probability $P(0<X<δ)$ for all real $δ>0$ and all random variables $X$ with log-concave p.d.f.'s such that $EX=0$ and $EX^2=1$ is obtained.