Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

Large ball probabilities, Gaussian comparison and anti-concentration

Friedrich Götze
Alexey Naumov
Vladimir Spokoiny
And Vladimir Ulyanov


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.

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