A GAUSSIAN SMALL DEVIATION INEQUALITY FOR CONVEX FUNCTIONS

• 成果类型: Article
• 署名作者: Paouris, Grigoris; Valettas, Petros
• 署名单位: Texas A&M University System; Texas A&M University College Station; University of Missouri System; University of Missouri Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1206
• 发表日期: 2018
• 页码: 1441-1454
• 关键词: brunn-minkowski inequality; Small Ball Probability; dvoretzkys theorem; random version

摘要

Let Z be an n-dimensional Gaussian vector and let f : R-n -> R be a convex function. We prove that P(f (Z) <= Ef (Z) - t root Var f(Z) <= = exp(-ct(2)), for all t > 1 where c > 0 is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.