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Section: New Results

Statistical aspects of Determinantal Point Processes

Participant : Frédéric Lavancier.

Determinantal point processes (DPPs) have been introduced in their general form by Macchi (1975) and have been extensively studied from a probabilistic point of view in the 2000’s (one of the main reason being their central role in random matrix theory). In [23] , we have demonstrated that DPPs provide useful models for the description of spatial point pattern datasets where nearby points repel each other. We have exploited the appealing probabilistic properties of DPPs to develop parametric models, where the likelihood and moment expressions can be easily evaluated and realizations can be quickly simulated. We have discussed how statistical inference is conducted using the likelihood or moment properties of DPP models, and we provided freely available software for simulation and statistical inference.

In [13] , we have addressed the question of how repulsive a stationary DPP can be, in order to assess the range of practical situations this promising class of models may model. We determine the most repulsive DPP (in some sense) and we introduce 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. Some theoretical aspects of inference for stationary DPPs are tackled in [37] and [38] . In the former study we have established the Brillinger mixing property of stationary DPPs, a first important step toward asymptotic inference. In the latter contribution, we have exploited this result to deduce the consistency and asymptotic properties of contrast estimators for stationary DPPs.

References:  [23] , [13] , [37] , [38]

Collaborators: Christophe Ange Napoléon Biscio (LMJL, University of Nantes),

                          Jesper Møller (Department of Mathematical Sciences, Aalborg University, Denmark),

                          Ege Rubak (Department of Mathematical Sciences, Aalborg University, Denmark).