论文标题
在紧凑型组II上随机步行的等分。 Wasserstein指标
Equidistribution of random walks on compact groups II. The Wasserstein metric
论文作者
论文摘要
我们考虑随机步行$ s_k $与I.I.D.配备双重变量度量的紧凑型组上的步骤。我们证明了总和$ \ sum_ {k = 1}^n f（s_k）$具有hölder连续测试函数$ f $的定量词法定理，包括中心限制定理，迭代次数的定律，几乎可以确定Wiener过程的近似值，并提供了$ s_k $ s_k $ segress $ s_ $ nmer $ segers $ se $ n $ n $ n $ n $ n $ n $ n.作为一个例子，我们构建离散的随机行走在圆环$ \ mathbb {r}^d/\ mathbb {z}^d $上的非理性晶格上，并在$ p $ -P $ -WASSERSTEANTEN中找到其确切的收敛速度。该证明使用了新的浆果 - 对于$ p $ wesserstein the the the the the the the the the the the the the the the the the the the lattice的不平等现象。这些结果与随机步行的第一部分相辅相成，具有绝对连续的成分和定量的千古定理，用于鲍尔可测量的测试函数。
We consider a random walk $S_k$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum $\sum_{k=1}^N f(S_k)$ 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 the distribution of $S_k$ 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 $\mathbb{R}^d/\mathbb{Z}^d$, 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.
