Positively (resp. negatively) associated point processes are a class of point processes that induce attraction (resp. inhibition) between the points. As an important example, determinantal point processes (DPPs) are negatively associated. We prove α-mixing properties for associated spatial point processes by controlling their α-coefficients in terms of the first two intensity functions. A central limit theorem for functionals of associated point processes is deduced, using both the association and the α-mixing properties. We discuss in detail the case of DPPs, for which we obtain the limiting distribution of sums, over subsets of close enough points of the process, of any bounded function of the DPP. As an application, we get the asymptotic properties of the parametric two-step estimator of some inhomogeneous DPPs.We then establish in Section 3 a general central limit theorem (CLT) for random fields defined as a function of an associated point process (Theorem 3.1). A standard method for proving this kind of theorem is to rely on sufficiently fast decaying α-mixing coefficients along with some moment assumptions. We use an alternative procedure that exploits both the mixing properties and the association property. This results in weaker assumptions on the underlying point process, that can have slower decaying mixing coefficients. This improvement allows in particular to include all standard DPPs, some of them being otherwise excluded with the first approach (like for instance DPPs associated to the Bessel-type kernels [4]).Section 4 discusses in detail the case of DPPs, where we derive a tight explicit bound for their α-mixing coefficients and prove a central limit theorem for certain functionals of a DPP (Theorem 4.4). Specifically, these functionals write as a sum of a bounded function of the DPP, over subsets of close-enough points of the DPP. A particular case concerns sums over p-tuple of close enough points of the DPP, which are frequently encountered in asymptotic inference. Limit theorems in this setting have been established in [38] when p " 1, and in [3] for stationary DPPs and p ě 1. We thus extend these studies to sums over any subsets and without the stationary assumption. As a statistical application, we consider the parametric estimation of second-order intensity reweighted stationary DPPs. These DPPs have an inhomogeneous first order intensity, but translation-invariant higher order (reweighted) intensities. We prove that the two-step estimator introduced in [39], designed for this kind of inhomogeneous point process models, is consistent and asymptotically normal when applied to DPPs. Associated point processes and α-mixing NotationIn this paper, we consider locally finite simple point processes on R d , for a fixed d P N. Some theoretical background on point processes can be found in [12,34]. We denote by Ω the set of locally finite point configurations in R d . For X P Ω and A Ă R d , we write N pAq :" cardpX ∩ Aq for the random variable representing the number of points of X that fall in A. W...

Estimating function inference is indispensable for many common point process models where the joint intensities are tractable while the likelihood function is not. In this paper we establish asymptotic normality of estimating function estimators in a very general setting of non-stationary point processes. We then adapt this result to the case of non-stationary determinantal point processes which are an important class of models for repulsive point patterns. In practice often first and second order estimating functions are used. For the latter it is common practice to omit contributions for pairs of points separated by a distance larger than some truncation distance which is usually specified in an ad hoc manner. We suggest instead a data-driven approach where the truncation distance is adapted automatically to the point process being fitted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies for non-stationary determinantal point processes.

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