论文标题
在圆上随机行走和二磷抗菌近似
Random walks on the circle and Diophantine approximation
论文作者
论文摘要
随机步行在圆圈组$ \ mathbb {r}/\ mathbb {z} $中，其基本步骤是晶格变量，带有span $α\ not \ in \ mathbb {q} $ {q} $或$ p/q \ in \ mathbb {q} $ in \ m mathbb {q} $采取mod $ \ m \ mathbb {z z} $ restion Inspory。在合理的情况下，我们在有限的环状亚组$ \ mathbb {z} _q $上随机步行，中央限制定理和迭代对数定律从有限的状态太空链中的经典结果遵循。在本文中，我们将这些结果扩展到不合理的跨度$α$的随机步行，并明确描述了这些马尔可夫链从有限状态到一般状态空间的过渡，为$ p/q \ to to p/q \ toα$，沿最佳理性近似值。我们还考虑了弱收敛到Kolmogorov指标中固定分布的速率，在合理的情况下，观察到从多项式到指数衰减的惊人过渡，大约q^2 $ steps；这似乎是紧凑型群体随机步行理论中的一种新现象。相反，在总变异度量中弱收敛到固定分布的速率纯粹是指数的。
Random walks on the circle group $\mathbb{R}/\mathbb{Z}$ whose elementary steps are lattice variables with span $α\not\in \mathbb{Q}$ or $p/q \in \mathbb{Q}$ taken mod $\mathbb{Z}$ exhibit delicate behavior. In the rational case we have a random walk on the finite cyclic subgroup $\mathbb{Z}_q$, and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper we extend these results to random walks with irrational span $α$, and explicitly describe the transition of these Markov chains from finite to general state space as $p/q \to α$ along the sequence of best rational approximations. We also consider the rate of weak convergence to the stationary distribution in the Kolmogorov metric, and in the rational case observe a surprising transition from polynomial to exponential decay after $\approx q^2$ steps; this seems to be a new phenomenon in the theory of random walks on compact groups. In contrast, the rate of weak convergence to the stationary distribution in the total variation metric is purely exponential.
