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Contribution to the modelling and the parametric estimation of determinantal point processes

This manuscript is devoted to the study of determinantal point processes (DPPs) and their parametric estimation. These processes are known to be well adapted to inhibitive point patterns, where the points tend to repel each others. In the first chapter, we study the flexibility of this model by suggesting two definitions of the repulsiveness, both based on the second order moments of the process, namely the pair correlation function (pcf) g. This leads us to identify the most repulsive DPP. Then, we introduce new parametric families of DPPs that cover a large range of DPPs, from the 
stationary Poisson process to the most repulsive DPP. In the second chapter, we prove that stationary DPPs are Brillinger mixing. This property allows us to deduce asymptotic results on several statistics. Namely, we prove a central limit theorem for a wide class of functionals of order p of the process and we adapt to DPPs some results already known on kernel estimators of the function g of Brillinger mixing point processes. Finally, in the last chapter we study the minimum contrast estimation based on the Ripley’s K-function and the pcf g. We prove the consistency and the asymptotic normality of these methods for stationary DPPs. In particular, we obtain an explicit form of the asymptotic variance. These results are in fact particular cases of a more general theorem dealing with the asymptotic properties of minimum contrast estimation for stationary point processes that we state and prove in this chapter.