An Indicator Function Limit Theorem in Dynamical Systems

TL;DR

This paper proves that in all aperiodic dynamical systems, one can construct sets whose normalized partial sums have distribution sequences dense in all probability measures, revealing a rich limit behavior.

Contribution

It provides a constructive proof demonstrating the existence of sets with dense distribution sequences in all aperiodic dynamical systems, extending understanding of limit theorems.

Findings

Existence of sets with dense distribution sequences in aperiodic systems

Constructive proof method for the indicator function limit theorem

Extension of limit theorems to a broad class of dynamical systems

Abstract

We show by a constructive proof that in all aperiodic dynamical system, for all sequences $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a set $A\in\A$ having the property that the sequence of the distributions of $(\frac{1}{a_{n}}S_{n}(\ind_A-\mu(A)))_{n\in\N}$ is dense in the space of all probability measures on $\R$.