Summary: A method is proposed for estimating the covariance structure of a nonparametric estimator for the reduced second moment measure, \(K(s)\), of a homogeneous planar Poisson process. The method relies on the invariance of the reduced second moment measure to random thinning, and the known covariance structure of the estimator under random sampling from a fixed set of points. The possible extension of the method to stationary Cox processes is discussed.