Statistical methodology for spatio-temporal point processes is in its infancy. We consider second-order analysis based on pair correlation functions and $K$-functions for first general inhomogeneous spatio-temporal point processes and second inhomogeneous spatio-temporal Cox processes. Assuming spatio-temporal separability of the intensity function, we clarify different meanings of second-order spatio-temporal separability. One is second-order spatio-temporal independence and relates e.g. to log-Gaussian Cox processes with an additive covariance structure of the underlying spatio-temporal Gaussian process. Another concerns shot-noise Cox processes with a separable spatio-temporal covariance density. We propose diagnostic procedures for checking hypotheses of second-order spatio-temporal separability, which we apply on simulated and real data (the UK 2001 epidemic foot and mouth disease data).

Key words: spatio-temporal functional summary statistics; K-function; pair correlation function; second-order intensity-reweighted stationarity; shot-noise Cox process; spatio-temporal separability