论文标题
一些反示例到中央限制定理以进行随机旋转
Some counterexamples to the central limit theorem for random rotations
论文作者
论文摘要
修复一个非理性数字$α$，并考虑在圆圈上随机步行，在该圆圈中，每个步骤都会移至$ x+α$或$ x-α$，概率为$ 1/2，1/2 $，提供当前位置为$ x $。如果给出了可观察的，我们可以研究一个随机步行的称为加性功能的过程。一个人可以在可观察到的$α$的可观察到的二聚体特性之间的某些关系中表示中心限制定理。在这里证明，对于每个liouville角度来说，都存在一个平稳的观察力，使中心限制定理失败。我们还构建了一个liouville角度，因此中心极限定理在某些可观察到的分析性上失败。对于Diophantine角度，也给出了一些反例。一个有趣的问题仍然开放。
Fix an irrational number $α$, and consider a random walk on the circle in which at each step one moves to $x+α$ or $x-α$ with probabilities $1/2, 1/2$ provided the current position is $x$. If an observable is given we can study a process called an additive functional of this random walk. One can formulate certain relations between the regularity of the observable and the Diophantine properties of $α$ implying the central limit theorem. It is proven here that for every Liouville angle there exists a smooth observable such that the central limit theorem fails. We construct also a Liouville angle such that the central limit theorem fails with some analytic observable. For Diophantine angles some counterexample is given as well. An interesting question remained open.
