Some counterexamples to the central limit theorem for random rotations

TL;DR

This paper constructs specific examples showing the central limit theorem fails for certain irrational rotations on the circle, especially for Liouville angles, highlighting limitations of classical probabilistic results in these settings.

Contribution

It provides explicit counterexamples of the central limit theorem failure for Liouville and some Diophantine angles, using smooth and analytic observables.

Findings

Counterexamples for Liouville angles with smooth observables

Counterexamples for Liouville angles with analytic observables

Counterexamples for some Diophantine angles

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.