We establish a central limit theorem for multivariate summary statistics of non-stationary $\alpha$-mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed e.g. to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.

Keywords: $\alpha$-mixing, central limit theorem, estimating function, random field, spatial point process, subsampling, summary statistics.