Gaussian mixtures: entropy and geometric inequalities

TL;DR

This paper studies Gaussian mixtures, deriving sharp inequalities for moments, entropy, and geometric measures, and extends classical Gaussian inequalities to this broader class of distributions.

Contribution

It introduces new sharp inequalities and extensions of classical Gaussian inequalities specifically for Gaussian mixtures, including entropy, moment, and geometric inequalities.

Findings

Sharp moment and entropy comparison estimates for Gaussian mixtures

Extensions of B-inequality and Gaussian correlation inequality to Gaussian mixtures

Sharp constants in Khintchine inequalities for p-norm unit balls

Abstract

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 Khintchine 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.