Asymptotic behaviour of discrepancy measures of second-order characteristics of spatial point processes with applications to model identification.

The aim of this project is the investigation of the asymptotic behaviour of distance measures of product densities, pair correlation functions and other second order characteristics and their nonparametric estimators in the setting of stationary (and isotropic) point processes in R^d. Starting from the assumption that one realization of a point field is observed in a convex window expanding in every direction, approved empirical second order quantities ( kernel-type product density estimators, Horvitz-Thompson estimator and others) and their distance to corresponding hypothetical second order quantities (measured, for instance, by the integrated squared error) are determined. Assuming that the hypothetical point process satisfies certain mixing conditions, (functional) central limit theorems are derived such that the asymptotic distribution depends only on the second order characteristic employed. These central limit theorems will result in ancillary goodness-of-fit tests. We plan to extend this approach to other stationary random measures such as fiber processes. The power of the suggested test procedures will be investigated by simulation studies, where particular interest lies in moderate window sizes. The test procedures will be implemented in the statistics software R.