Open Access
September 2018 Gaussian mixtures: Entropy and geometric inequalities
Alexandros Eskenazis, Piotr Nayar, Tomasz Tkocz
Ann. Probab. 46(5): 2908-2945 (September 2018). DOI: 10.1214/17-AOP1242


A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to $e^{-|t|^{p}}$ and symmetric $p$-stable random variables, where $p\in(0,2]$. We obtain various sharp moment and entropy comparison estimates for weighted sums of independent Gaussian mixtures and investigate extensions of the B-inequality and the Gaussian correlation inequality in the context of Gaussian mixtures. We also obtain a correlation inequality for symmetric geodesically convex sets in the unit sphere equipped with the normalized surface area measure. We then apply these results to derive sharp constants in Khinchine inequalities for vectors uniformly distributed on the unit balls with respect to $p$-norms and provide short proofs to new and old comparison estimates for geometric parameters of sections and projections of such balls.


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Alexandros Eskenazis. Piotr Nayar. Tomasz Tkocz. "Gaussian mixtures: Entropy and geometric inequalities." Ann. Probab. 46 (5) 2908 - 2945, September 2018.


Received: 1 November 2016; Revised: 1 October 2017; Published: September 2018
First available in Project Euclid: 24 August 2018

zbMATH: 06964351
MathSciNet: MR3846841
Digital Object Identifier: 10.1214/17-AOP1242

Primary: 60E15
Secondary: 52A20 , 52A40 , 94A17

Keywords: B-inequality , Correlation inequalities , Entropy , extremal sections and projections of $\ell_{p}$-balls , Gaussian measure , Gaussian mixture , Khinchine inequality , Small ball probability

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 5 • September 2018
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