Berry–Esseen Smoothing Inequality for the Wasserstein Metric on Compact Lie Groups

Borda, Bence (2021) Berry–Esseen Smoothing Inequality for the Wasserstein Metric on Compact Lie Groups. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS. pp. 1-24. ISSN 1069-5869 (print); 1531-5851 (online)

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Abstract

We prove a sharp general inequality estimating the distance of two probability measures on a compact Lie group in the Wasserstein metric in terms of their Fourier transforms. We use a generalized form of the Wasserstein metric, related by Kantorovich duality to the family of functions with an arbitrarily prescribed modulus of continuity. The proof is based on smoothing with a suitable kernel, and a Fourier decay estimate for continuous functions. As a corollary, we show that the rate of convergence of random walks on semisimple groups in the Wasserstein metric is necessarily almost exponential, even without assuming a spectral gap. Applications to equidistribution and empirical measures are also given.

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