Large ball probabilities, Gaussian comparison and anti-concentration - Friedrich Götze; Alexey Naumov; Vladimir Spokoiny; Vladimir Ulyanov (Humboldt-Universität zu Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series", 2018)
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: C00; 330; Gaussian comparison; Gaussian anti-concentration inequalities; effective rank; dimension free bounds; Schatten norm; high-dimensional inference