A Gaussian small deviation inequality for convex functions

Abstract

Let $Z$ be an $n$-dimensional Gaussian vector and let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a convex function. We show that: $$\mathbb{P}\left(f(Z)\le \mathbb{E}f(Z)-t\sqrt{\mathrm{Var}f(Z)}\right)\le \exp (-c{t}^2),$$ for all $t>1$, where $c>0$ is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.

14 pages; major revision, to appear in Ann. Prob