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Gaussian mixtures: entropy and geometric inequalities. (English) Zbl 1428.60036
Summary: 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.

MSC:
60E15 Inequalities; stochastic orderings
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A40 Inequalities and extremum problems involving convexity in convex geometry
94A17 Measures of information, entropy
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