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Götze, Friedrich; Naumov, Alexey; Spokoiny, Vladimir; Ulyanov, Vladimir

Large ball probabilities, Gaussian comparison and anti-concentration. (English) Zbl 1428.62187

Bernoulli 25, No. 4A, 2538-2563 (2019).
Summary: 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.

Cited in 14 Documents

MSC:

60B11 Probability theory on linear topological spaces
60E15 Inequalities; stochastic orderings
60F05 Central limit and other weak theorems

Keywords:

dimension free bounds; Gaussian anti-concentration inequalities; Gaussian comparison; high-dimensional CLT; high-dimensional inference; Schatten norm
PDFBibTeX XMLCite
\textit{F. Götze} et al., Bernoulli 25, No. 4A, 2538--2563 (2019; Zbl 1428.62187)
Full Text: DOI arXiv Euclid

References:

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