Equidistribution of random walks on compact groups II. The Wasserstein metric

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