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arXiv · 2005.04925

Berry-Esseen smoothing inequality for the Wasserstein metric on compact Lie groups

Bence Borda
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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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Bence Borda. 2020-05-11. Berry-Esseen smoothing inequality for the Wasserstein metric on compact Lie groups. https://arxiv.org/abs/2005.04925

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