Limit theory for geometric statistics of point processes having fast decay of correlations

Let $P$ be a simple,stationary point process having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $P_n:= P \cap W_n$ be its restriction to windows $W_n:= [-{1 \over 2}n^{1/d},{1 \over 2}n^{1/d}]^d \subset \mathbb{R}^d$. We consider the statistic $H_n^\xi:= \sum_{x \in P_n}\xi(x,P_n)$ where $\xi(x,P_n)$ denotes a score function representing the interaction of $x$ with respect to $P_n$. When $\xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and CLT for $H_n^{\xi}$ and, more generally, for statistics of the re-scaled, possibly signed, $\xi$-weighted point measures $\mu_n^{\xi} := \sum_{x \in P_n} \xi(x,P_n) \delta_{n^{-1/d}x}$, as $W_n \uparrow \mathbb{R}^d$. This gives the limit theory for non-linear geometric statistics (such as clique counts, intrinsic volumes of the Boolean model, and total edge length of the $k$-nearest neighbors graph) of $\alpha$-determinantal point processes having fast decreasing kernels extending the CLTs of Soshnikov (2002) to non-linear statistics. It also gives the limit theory for geometric U-statistics of $\alpha$-permanental point processes and the zero set of Gaussian entire functions, extending the CLTs of Nazarov and Sodin (2012) and Shirai and Takahashi (2003), which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in Blaszczyszyn (1995), Blaszczyszyn, Merzbach, Schmidt (1997) to show the fast decay of the correlations of $\xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the CLT for $\mu_n^\xi$ via an extension of the cumulant method.