The Annals of Probability

Gaussian mixtures: Entropy and geometric inequalities

Alexandros Eskenazis, Piotr Nayar, and Tomasz Tkocz

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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.

Article information

Ann. Probab., Volume 46, Number 5 (2018), 2908-2945.

Received: November 2016
Revised: October 2017
First available in Project Euclid: 24 August 2018

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 52A40: Inequalities and extremum problems 94A17: Measures of information, entropy

Gaussian measure Gaussian mixture Khinchine inequality entropy B-inequality small ball probability correlation inequalities extremal sections and projections of $\ell_{p}$-balls


Eskenazis, Alexandros; Nayar, Piotr; Tkocz, Tomasz. Gaussian mixtures: Entropy and geometric inequalities. Ann. Probab. 46 (2018), no. 5, 2908--2945. doi:10.1214/17-AOP1242.

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