## Bernoulli

- Bernoulli
- Volume 25, Number 4B (2019), 3555-3589.

### Asymptotic equivalence of fixed-size and varying-size determinantal point processes

Simon Barthelmé, Pierre-Olivier Amblard, and Nicolas Tremblay

#### Abstract

Determinantal Point Processes (DPPs) are popular models for point processes with repulsion. They appear in numerous contexts, from physics to graph theory, and display appealing theoretical properties. On the more practical side of things, since DPPs tend to select sets of points that are some distance apart (repulsion), they have been advocated as a way of producing random subsets with high diversity. DPPs come in two variants: fixed-size and varying-size. A sample from a varying-size DPP is a subset of random cardinality, while in fixed-size “$k$-DPPs” the cardinality is fixed. The latter makes more sense in many applications, but unfortunately their computational properties are less attractive, since, among other things, inclusion probabilities are harder to compute. In this work, we show that as the size of the ground set grows, $k$-DPPs and DPPs become equivalent, in the sense that fixed-order inclusion probabilities converge. As a by-product, we obtain saddlepoint formulas for inclusion probabilities in $k$-DPPs. These turn out to be extremely accurate, and suffer less from numerical difficulties than exact methods do. Our results also suggest that $k$-DPPs and DPPs also have equivalent maximum likelihood estimators. Finally, we obtain results on asymptotic approximations of elementary symmetric polynomials which may be of independent interest.

#### Article information

**Source**

Bernoulli, Volume 25, Number 4B (2019), 3555-3589.

**Dates**

Received: March 2018

Revised: August 2018

First available in Project Euclid: 25 September 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1569398777

**Digital Object Identifier**

doi:10.3150/18-BEJ1102

**Mathematical Reviews number (MathSciNet)**

MR4010965

**Zentralblatt MATH identifier**

07110148

**Keywords**

determinantal point processes point processes saddlepoint approximation

#### Citation

Barthelmé, Simon; Amblard, Pierre-Olivier; Tremblay, Nicolas. Asymptotic equivalence of fixed-size and varying-size determinantal point processes. Bernoulli 25 (2019), no. 4B, 3555--3589. doi:10.3150/18-BEJ1102. https://projecteuclid.org/euclid.bj/1569398777