Number of visits in arbitrary sets for $\phi$-mixing dynamics

TL;DR

This paper extends Poisson approximation results for visit counts in mixing dynamical systems to arbitrary sets with vanishing measure, using Stein-Chen method and providing applications to $g$-measures.

Contribution

It generalizes visit distribution approximations to arbitrary sets in $\,phi$-mixing systems, beyond traditional balls and cylinders.

Findings

Poisson approximation with explicit error bounds in total variation

Applicability to arbitrary sets with vanishing measure

Applications to temporal synchronization of $g$-measures

Abstract

It is well-known that, for sufficiently mixing dynamical systems, the number of visits to balls and cylinders of vanishing measure is approximately Poisson compound distributed in the Kac scaling. Here we extend this kind of results when the target set is an arbitrary set with vanishing measure in the case of $\phi$-mixing systems. The error of approximation in total variation is derived using Stein-Chen method. An important part of the paper is dedicated to examples to illustrate the assumptions, as well as applications to temporal synchronisation of $g$-measures