Some Counterexamples to the Central Limit Theorem for Random Rotations

Abstract

Fix an irrational number alpha , and consider a random walk on the circle in which at each step one moves to x+alpha or x-alpha 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 alpha 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.