## Bernoulli

• Bernoulli
• Volume 24, Number 2 (2018), 1171-1201.

### Determinantal point process models on the sphere

#### Abstract

We consider determinantal point processes on the $d$-dimensional unit sphere $\mathbb{S}^{d}$. These are finite point processes exhibiting repulsiveness and with moment properties determined by a certain determinant whose entries are specified by a so-called kernel which we assume is a complex covariance function defined on $\mathbb{S}^{d}\times\mathbb{S}^{d}$. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. Particularly, we characterize and construct isotropic DPPs models on $\mathbb{S}^{d}$, where it becomes essential to specify the eigenvalues and eigenfunctions in a spectral representation for the kernel, and we figure out how repulsive isotropic DPPs can be. Moreover, we discuss the shortcomings of adapting existing models for isotropic covariance functions and consider strategies for developing new models, including a useful spectral approach.

#### Article information

Source
Bernoulli, Volume 24, Number 2 (2018), 1171-1201.

Dates
Revised: July 2016
First available in Project Euclid: 21 September 2017

https://projecteuclid.org/euclid.bj/1505980893

Digital Object Identifier
doi:10.3150/16-BEJ896

Mathematical Reviews number (MathSciNet)
MR3706791

Zentralblatt MATH identifier
06778362

#### Citation

Møller, Jesper; Nielsen, Morten; Porcu, Emilio; Rubak, Ege. Determinantal point process models on the sphere. Bernoulli 24 (2018), no. 2, 1171--1201. doi:10.3150/16-BEJ896. https://projecteuclid.org/euclid.bj/1505980893

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