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Log-Concavity and Strong Log-Concavity: a review.

Adrien Saumard¹, Jon A Wellner²

¹Departamento de Estadística, CIMFAV, Universidad de Valparaíso, Chile.

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Summary

This summary is machine-generated.

This study explores log-concavity and strong log-concavity, proving their preservation under convolution using Efron's monotonicity theorem. A new proof employs the asymmetric Brascamp-Lieb inequality, linking these concepts to broader mathematical fields.

Keywords:

concave convex convolution inequalities log-concave monotone preservation strong log-concave

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Area of Science:

Mathematics and Statistics
Probability Theory
Convex Geometry

Background:

Log-concavity and strong log-concavity are crucial properties in probability and statistics.
Understanding their behavior under operations like convolution is essential for theoretical advancements.

Purpose of the Study:

To review and formulate results on log-concavity and strong log-concavity in discrete and continuous settings.
To demonstrate the preservation of these properties under convolution.
To establish connections with other mathematical and statistical areas.

Main Methods:

Review of existing literature on log-concavity.
Application of Efron's (1969) monotonicity theorem.
Development of a new proof for Efron's theorem utilizing the asymmetric Brascamp-Lieb inequality (Otto and Menz, 2013).

Main Results:

Log-concavity and strong log-concavity are preserved under convolution on ℝ.
A novel proof of Efron's theorem is presented.
Connections are established between log-concavity and concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

Conclusions:

The preservation of log-concavity under convolution is a fundamental result with broad implications.
The study provides new insights and a unified perspective on log-concavity.
This work bridges concepts across various mathematical and statistical disciplines.