Open Access
May 2018 A Gaussian small deviation inequality for convex functions
Grigoris Paouris, Petros Valettas
Ann. Probab. 46(3): 1441-1454 (May 2018). DOI: 10.1214/17-AOP1206


Let $Z$ be an $n$-dimensional Gaussian vector and let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a convex function. We prove that

\[\mathbb{P}(f(Z)\leq \mathbb{E}f(Z)-t\sqrt{\operatorname{Var}f(Z)})\leq\exp (-ct^{2}),\] for all $t>1$ where $c>0$ is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.


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Grigoris Paouris. Petros Valettas. "A Gaussian small deviation inequality for convex functions." Ann. Probab. 46 (3) 1441 - 1454, May 2018.


Received: 1 November 2016; Revised: 1 May 2017; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894778
MathSciNet: MR3785592
Digital Object Identifier: 10.1214/17-AOP1206

Primary: 60D05
Secondary: 52A21 , 52A23

Keywords: concentration for convex functions , Ehrhard’s inequality , Johnson–Lindenstrauss lemma , Small ball probability

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 3 • May 2018
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