Abstract
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.
Citation
Grigoris Paouris. Petros Valettas. "A Gaussian small deviation inequality for convex functions." Ann. Probab. 46 (3) 1441 - 1454, May 2018. https://doi.org/10.1214/17-AOP1206
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