Gaussian limits of empirical multiparameter K-functions of homogeneous Poisson processes and tests for complete spatial randomness

We prove two functional limit theorems for empirical multiparameter second moment functions (generalizing Ripley’s K-function) obtained from a homogeneous Poisson point field observed in an unboundedly expanding convex sampling window Wn in ℝd. The cases of known and unknown (estimated) intensity lead to distinct Gaussian limits and require quite different proofs. Further we determine the limit distributions of the maximal deviation and the integrated squared distance between empirical and true multiparameter second moment function. These results give rise to construct goodness-of-fit tests for checking the hypothesis that a given point pattern is completely spatially random (CSR), that is, a realization of a homogeneous Poisson process.