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

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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.

Original languageEnglish
Article number13
Number of pages23
JournalThe Journal of Fourier Analysis and Applications
Issue number2
Publication statusPublished - Apr 2021


  • Compact group
  • Equidistribution
  • Erdős–Turán inequality
  • Fourier transform
  • Random walk
  • Spectral gap
  • Transport metric

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Mathematics(all)

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