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

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.


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
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