Large ball probability, Gaussian comparison and anti-concentration

TL;DR

This paper provides dimension-free, tight bounds for the probability differences of Gaussian elements hitting a ball in a Hilbert space, with applications to statistical inference and high-dimensional CLT.

Contribution

It introduces novel non-asymptotic bounds based on the nuclear norm of covariance differences, improving upon previous inequalities like Pinsker's.

Findings

Bounds are dimension-free and depend on nuclear norm and mean shift.

Significant improvement over Pinsker's inequality-based bounds.

Applications demonstrated in statistical inference and high-dimensional CLT.

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.