Geometric statistics of 'clustering' point sets

Abstract:   Many statistics of Euclidean point sets P (i.e., locally finite counting measures on R^d) are expressed as a sum of spatially dependent terms. Examples of such statistics include clique counts of a random geometric graph on P, edge-lengths of the k-nearest neighbour graph on P and intrinsic volumes of unions of balls centered at P.  A rich theory exists for aforementioned statistics when the point set P consists of 'independently' distributed points i.e., a Poisson point process. Here, we establish a limit theory - expectation and variance asymptotics as well as a central limit theorem - when the point set P consists of points distributed 'dependently' but having 'asymptotic independence'. We precisely formulate 'asymptotic independence' via the notion of 'clustering' arising in statistical physics. The assumption of clustering holds for various point processes such as Gibbs point processes, determinantal point processes, permanental point processes and zeros of Gaussian entire functions.  As a consequence of our general theory, we can derive limit theorems for the statistics mentioned above when the underlying point set P is one of the above point