Open Access
November 2019 Large ball probabilities, Gaussian comparison and anti-concentration
Friedrich Götze, Alexey Naumov, Vladimir Spokoiny, Vladimir Ulyanov
Bernoulli 25(4A): 2538-2563 (November 2019). DOI: 10.3150/18-BEJ1062


We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker’s inequality via the Kullback–Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in Hilbert space. The paper presents a number of examples motivating our results and applications of the obtained bounds to statistical inference and to high-dimensional CLT.


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Friedrich Götze. Alexey Naumov. Vladimir Spokoiny. Vladimir Ulyanov. "Large ball probabilities, Gaussian comparison and anti-concentration." Bernoulli 25 (4A) 2538 - 2563, November 2019.


Received: 1 March 2018; Revised: 1 August 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110104
MathSciNet: MR4003557
Digital Object Identifier: 10.3150/18-BEJ1062

Keywords: dimension free bounds , Gaussian anti-concentration inequalities , Gaussian comparison , high-dimensional CLT , high-dimensional inference , Schatten norm

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4A • November 2019
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