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 the Pinsker inequality via the Kullback–Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in a Hilbert space. A number of examples are also provided, motivating the results and its applications to statistical inference and high-dimensional CLT.
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Fujikoshi, Y., Ulyanov, V.V. (2020). Gaussian Comparison and Anti-concentration. In: Non-Asymptotic Analysis of Approximations for Multivariate Statistics. SpringerBriefs in Statistics(). Springer, Singapore. https://doi.org/10.1007/978-981-13-2616-5_8
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DOI: https://doi.org/10.1007/978-981-13-2616-5_8
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