A Gaussian small deviation inequality for convex functions

TL;DR

This paper establishes a Gaussian small deviation inequality for convex functions, providing bounds on the probability that such functions deviate below their mean by a certain amount, with applications to Gaussian processes.

Contribution

It introduces a new inequality for convex functions of Gaussian vectors, extending small deviation results with variance-sensitive bounds.

Findings

Proves a Gaussian small deviation inequality for convex functions.

Derives variance-sensitive small ball probabilities for Gaussian processes.

Provides explicit exponential decay bounds for deviations.

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