TY - JOUR
T1 - Equidistribution of random walks on compact groups II. The Wasserstein metric
AU - Borda, Bence
N1 - Publisher Copyright:
© 2021 International Statistical Institute. All rights reserved.
PY - 2021/11
Y1 - 2021/11
N2 - 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.
AB - 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.
KW - Berry-Esseen inequality
KW - Central limit theorem
KW - Empirical distribution
KW - Ergodic theorem
KW - Law of the iterated logarithm
KW - Simultaneous Diophantine approximation
UR - http://www.scopus.com/inward/record.url?scp=85114722519&partnerID=8YFLogxK
U2 - 10.3150/21-BEJ1324
DO - 10.3150/21-BEJ1324
M3 - Article
SN - 1350-7265
VL - 27
SP - 2598
EP - 2623
JO - Bernoulli
JF - Bernoulli
IS - 4
ER -