Short Communications

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

M. Wang

School of Mathematics and Statistics, Wuhan University, Wuhan, China

Abstract: By the Prékopa–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.

Keywords: log-concavity, independent difference.

Received: 01.04.2024

DOI: 10.4213/tvp5715

 English version: Theory of Probability and its Applications, 2024, 69:3, 503–504

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