# Mixing properties and central limit theorem for associated point processes

## TL;DR

This paper establishes mixing properties and a central limit theorem for associated point processes, including determinantal point processes, enabling better understanding of their asymptotic behaviors and statistical inference methods.

## Contribution

It proves $ ext{α}$-mixing properties for associated point processes and derives a central limit theorem, with detailed analysis for determinantal point processes.

## Key findings

- $ ext{α}$-mixing coefficients are controlled by first two intensity functions.
- A central limit theorem is established for functionals of associated point processes.
- Asymptotic properties of parametric estimators for inhomogeneous DPPs are derived.

## Abstract

Positively (resp. negatively) associated point processes are a class of point processes that induce attraction (resp. inhibition) between the points. As an important example, determinantal point processes (DPPs) are negatively associated. We prove $\alpha$-mixing properties for associated spatial point processes by controlling their $\alpha$-coefficients in terms of the first two intensity functions. A central limit theorem for functionals of associated point processes is deduced, using both the association and the $\alpha$-mixing properties. We discuss in detail the case of DPPs, for which we obtain the limiting distribution of sums, over subsets of close enough points of the process, of any bounded function of the DPP. As an application, we get the asymptotic properties of the parametric two-step estimator of some inhomogeneous DPPs.