A Gaussian small deviation inequality for convex functions. (English) Zbl 1429.60022

Paouris, Grigoris; Valettas, Petros

Ann. Probab. 46, No. 3, 1441-1454 (2018).

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

MSC:

60D05	Geometric probability and stochastic geometry
52A21	Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
52A23	Asymptotic theory of convex bodies

Keywords:

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

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