Central limit theorem for the integrated squared error of the empirical second-order product density and goodness-of-fit tests for stationary point processes

Spatial point processes are mathematical models for irregular or random point patterns in the d-dimensional space, where usually d = 2 or d = 3 in applications. The second-order product density and its isotropic analogue, the pair correlation function, are important tools for analyzing stationary point processes. In the present work we derive central limit theorems for the integrated squared error (ISE) of the empirical second-order product density and for the ISE of the empirical pair correlation function when the observation window expands unboundedly. The proof techniques are based on higher-order cumulant measures and the Brillinger-mixing property of the underlying point processes. The obtained Gaussian limits are used to construct asymptotic goodness-of-fit tests for checking point process hypotheses even in the non-Poissonian case.