An example of a non-log-concave distribution where the difference has a log-concave density

TL;DR

This paper provides a counterexample showing that the difference of two i.i.d. random variables with log-concave densities does not necessarily have a log-concave density, challenging a common assumption.

Contribution

It constructs an explicit example demonstrating that the difference of i.i.d. log-concave variables may not be log-concave, disproving a potential converse to the Prékopa-Leindler inequality.

Findings

Counterexample where difference has a log-concave density

Disproves the converse of a known inequality

Highlights limitations of log-concavity properties

Abstract

By the Pr\'ekopa-Leindler inequality, the difference $X-X'$ has a log-concave density provided that $X$ has a log-concave density and $X, X'$ are independent and identically distributed. We prove that the opposite direction does not always hold true by giving an explicit example.