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arXiv · 1404.5886

Log-concavity and strong log-concavity: a review

Adrien Saumard·Jon A. Wellner
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Abstract

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

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Adrien Saumard, Jon A. Wellner. 2014-04-23. Log-concavity and strong log-concavity: a review. https://arxiv.org/abs/1404.5886

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