## Abstract

We consider a random walk Sk with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum _N k=1 f (Sk) with Hölder continuous test functions f , including the central limit theorem, the law of the iterated logarithm and an almost sure approximation by a Wiener process, provided that the distribution of Sk converges to the Haar measure in the p-Wasserstein metric fast enough. As an example, we construct discrete random walks on an irrational lattice on the torus Rd /Zd , and find their precise rate of convergence to uniformity in the p-Wasserstein metric. The proof uses a new Berry-Esseen type inequality for the p-Wasserstein metric on the torus, and the simultaneous Diophantine approximation properties of the lattice. These results complement the first part of this paper on random walks with an absolutely continuous component and quantitative ergodic theorems for Borel measurable test functions.

Original language | English |
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Pages (from-to) | 2598-2623 |

Number of pages | 26 |

Journal | Bernoulli |

Volume | 27 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 2021 |

## Keywords

- Berry-Esseen inequality
- Central limit theorem
- Empirical distribution
- Ergodic theorem
- Law of the iterated logarithm
- Simultaneous Diophantine approximation

## ASJC Scopus subject areas

- Statistics and Probability