Large ball probabilities, Gaussian comparison and anti-concentration

Friedrich Götze, Alexey Naumov, Vladimir Spokoiny and Vladimir Ulyanov

No 2018-026, IRTG 1792 Discussion Papers from Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series"

Abstract: 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.

Keywords: Gaussian comparison; Gaussian anti-concentration inequalities; effective rank; dimension free bounds; Schatten norm; high-dimensional inference (search for similar items in EconPapers)
JEL-codes: C00 (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations: View citations in EconPapers (18)

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Persistent link: https://EconPapers.repec.org/RePEc:zbw:irtgdp:2018026

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