GAUSSIAN MIXTURES: ENTROPY AND GEOMETRIC INEQUALITIES

• 成果类型: Article
• 署名作者: Eskenazis, Alexandros; Nayar, Piotr; Tkocz, Tomasz
• 署名单位: Princeton University; University of Warsaw
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1242
• 发表日期: 2018
• 页码: 2908-2945
• 关键词: small ball probability; convex-functions; b conjecture; unit ball; sections; constants; Majorization; variables; bodies; sums

摘要

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(-vertical bar t vertical bar P )and symmetric p-stable random variables, where p is an element of (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.