On Sums of Indicator Functions in Dynamical Systems

TL;DR

This paper investigates the limit behavior of sums of indicator functions in dynamical systems, showing that under certain conditions, the distributions of normalized partial sums can be dense in the space of probability measures.

Contribution

It demonstrates the existence of sets with dense distributional behavior of sums in aperiodic and ergodic dynamical systems, extending understanding of limit theorems in this context.

Findings

Distributions of normalized sums are dense in probability measures.

Existence of a dense G_delta set of such sets in ergodic systems.

Results hold for a wide class of aperiodic systems.

Abstract

In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every aperiodic dynamical system, for every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a measurable set $A$ such that the sequence of the distributions of the partial sums $\frac{1}{a_n}\sum_{i=0}^{n-1}(\ind_A-\mu(A))\circ T^i$ is dense in the set of the probability measures on $\R$. Further, in the ergodic case, we prove that there exists a dense $G_\delta$ of such sets.