## Statistics Surveys

### Log-concavity and strong log-concavity: A review

#### 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 strong log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1965). 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.

#### Article information

Source
Statist. Surv., Volume 8 (2014), 45-114.

Dates
First available in Project Euclid: 9 December 2014

https://projecteuclid.org/euclid.ssu/1418134163

Digital Object Identifier
doi:10.1214/14-SS107

Mathematical Reviews number (MathSciNet)
MR3290441

Zentralblatt MATH identifier
1360.62055

#### Citation

Saumard, Adrien; Wellner, Jon A. Log-concavity and strong log-concavity: A review. Statist. Surv. 8 (2014), 45--114. doi:10.1214/14-SS107. https://projecteuclid.org/euclid.ssu/1418134163

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