Brillinger mixing of determinantal point processes and statistical applications

Christophe A N Biscio,Frédéric Lavancier

doi:10.1214/16-ejs1116

Abstract

Stationary determinantal point processes are proved to be Brillinger mixing. This property is an important step towards asymptotic statistics for these processes. As an important example, a central limit theorem for a wide class of functionals of determinantal point processes is established. This result yields in particular the asymptotic normality of the estimator of the intensity of a stationary determinantal point process and of the kernel estimator of its pair correlation.

Highlights

Determinantal point processes (DPPs) are models for repulsive point patterns, where nearby points of the process tend to repel each other
DPPs have been applied in machine learning [20], spatial statistics [22, 21] and telecommunication [6, 24]
We focus in this paper on stationary DPPs on the continuous space Rd and we prove that they are Brillinger mixing

Summary

Introduction

Determinantal point processes (DPPs) are models for repulsive point patterns, where nearby points of the process tend to repel each other. The growing interest for DPPs in the statistical community is due to their appealing properties: They can be quickly and perfectly simulated, parametric models can be constructed, their moments are known and the likelihood has a closed form expression.

Determinantal point processes

Moment measures and Brillinger mixing

Main result

Statistical applications

Asymptotic behaviour of functionals of order p

Complement on the moments and cumulants of a point process