## Bernoulli

- Bernoulli
- Volume 25, Number 3 (2019), 1724-1754.

### Mixing properties and central limit theorem for associated point processes

Arnaud Poinas, Bernard Delyon, and Frédéric Lavancier

#### 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.

#### Article information

**Source**

Bernoulli, Volume 25, Number 3 (2019), 1724-1754.

**Dates**

Received: May 2017

Revised: February 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1560326425

**Digital Object Identifier**

doi:10.3150/18-BEJ1033

**Mathematical Reviews number (MathSciNet)**

MR3961228

**Zentralblatt MATH identifier**

07066237

**Keywords**

determinantal point process negative association parametric estimation positive association strong mixing

#### Citation

Poinas, Arnaud; Delyon, Bernard; Lavancier, Frédéric. Mixing properties and central limit theorem for associated point processes. Bernoulli 25 (2019), no. 3, 1724--1754. doi:10.3150/18-BEJ1033. https://projecteuclid.org/euclid.bj/1560326425