Weak dependence of point processes and application to second-order statistics

We propose a general definition for weak dependence of point processes as an alternative to mixing definitions. We give examples of such weak dependent point processes for the families of Neyman Scott processes or Cox processes. For these processes, we consider the empirical estimator of the empty space function. Using the general setting of the weak dependence property, we show the Central Limit Theorem for a vector of such statistics with different r. This completes results establishing the Central Limit Theorem under the Poisson process hypothesis.