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A Gaussian small deviation inequality for convex functions
Grigoris Paouris, Petros Valettas · 2016-11-06 · via math.PR updates on arXiv.org

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.

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