The Annals of Probability

A Gaussian small deviation inequality for convex functions

Grigoris Paouris and Petros Valettas

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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.

Article information

Ann. Probab., Volume 46, Number 3 (2018), 1441-1454.

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

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] 52A23: Asymptotic theory of convex bodies [See also 46B06]

Ehrhard’s inequality concentration for convex functions small ball probability Johnson–Lindenstrauss lemma


Paouris, Grigoris; Valettas, Petros. A Gaussian small deviation inequality for convex functions. Ann. Probab. 46 (2018), no. 3, 1441--1454. doi:10.1214/17-AOP1206.

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