Equivalence and orthogonality of Gaussian measures on spheres

The equivalence of Gaussian measures is a fundamental tool to establish the asymptotic properties of both prediction and estimation of Gaussian fields under fixed domain asymptotics. The paper solves Problem 18 in the list of open problems proposed by Gneiting (2013). Specifically, necessary and sufficient conditions are given for the equivalence of Gaussian measures associated to random fields defined on the d-dimensional sphere Sd, and with covariance functions depending on the great circle distance. We also focus on a comparison of our result with existing results on the equivalence of Gaussian measures for isotropic Gaussian fields on Rd+1 restricted to the sphere Sd. For such a case, the covariance function depends on the chordal distance being an approximation of the true distance between two points located on the sphere. Finally, we provide equivalence conditions for some parametric families of covariance functions depending on the great circle distance. An important implication of our results is that all the parameters indexing some families of covariance functions on spheres can be consistently estimated. A simulation study illustrates our findings in terms of implications on the consistency of the maximum likelihood estimator under fixed domain asymptotics.