One-sided concentration near the mean of log-concave distributions

TL;DR

This paper establishes a lower bound on the probability that a zero-mean, unit-variance log-concave distribution's variable falls within a small interval around zero, highlighting concentration near the mean.

Contribution

It provides a universal lower bound on the probability near the mean for all log-concave distributions with specified mean and variance.

Findings

Lower bound on P(0<X<δ) for all δ>0 and log-concave distributions

Results apply universally to all such distributions with given mean and variance

Enhances understanding of distribution concentration around the mean

Abstract

A lower bound on the probability $P(0<X<\delta)$ for all real $\delta>0$ and all random variables $X$ with log-concave p.d.f.'s such that $EX=0$ and $EX^2=1$ is obtained.