## Bernoulli

• Bernoulli
• Volume 22, Number 4 (2016), 2001-2028.

### Quantifying repulsiveness of determinantal point processes

#### Abstract

Determinantal point processes (DPPs) have recently proved to be a useful class of models in several areas of statistics, including spatial statistics, statistical learning and telecommunications networks. They are models for repulsive (or regular, or inhibitive) point processes, in the sense that nearby points of the process tend to repel each other. We consider two ways to quantify the repulsiveness of a point process, both based on its second-order properties, and we address the question of how repulsive a stationary DPP can be. We determine the most repulsive stationary DPP, when the intensity is fixed, and for a given $R>0$ we investigate repulsiveness in the subclass of $R$-dependent stationary DPPs, that is, stationary DPPs with $R$-compactly supported kernels. Finally, in both the general case and the $R$-dependent case, we present some new parametric families of stationary DPPs that can cover a large range of DPPs, from the stationary Poisson process (the case of no interaction) to the most repulsive DPP.

#### Article information

Source
Bernoulli, Volume 22, Number 4 (2016), 2001-2028.

Dates
Revised: February 2015
First available in Project Euclid: 3 May 2016

https://projecteuclid.org/euclid.bj/1462297673

Digital Object Identifier
doi:10.3150/15-BEJ718

Mathematical Reviews number (MathSciNet)
MR3498021

Zentralblatt MATH identifier
1343.60058

#### Citation

Biscio, Christophe Ange Napoléon; Lavancier, Frédéric. Quantifying repulsiveness of determinantal point processes. Bernoulli 22 (2016), no. 4, 2001--2028. doi:10.3150/15-BEJ718. https://projecteuclid.org/euclid.bj/1462297673

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#### Supplemental materials

• Supplement to “Quantifying repulsiveness of determinantal point processes”. We provide some illustrations of the nonuniqueness of the most globally repulsive DPP in the sense of Definition 1.1, as stated in Proposition 3.1. We also show the p.c.f.s and some realizations associated to different values of the parameters for the parametric families of DPPs introduced in Section 5.