Seminar at SMU Delhi
April 25, 2017 (Tuesday) ,
3:30 PM at Webinar
Geometric statistics of 'clustering' point sets
Abstract of Talk
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 afore-mentioned 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 processes.
Though there are other formulations of 'asymptotic independence' of random point sets, such formulations do not always yield a general limit theory covering all the examples of statistics as well as point processes mentioned above. This is a joint work with B. Blaszczyszyn and J. E. Yukich.