  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 $\alpha$-mixing properties for associated spatial point
processes by controlling their $\alpha$-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 $\alpha$-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.
