Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators

Abstract

We consider two Gaussian measures μ, ˜μ on a separable Hilbert space, with fractional-order covariance operators A−2β and Ã−2˜β, respectively, and derive necessary and sufficient conditions on A, à and β, ˜β > 0 for I. equivalence of the measures μ and ˜μ, and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure ˜μ. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle–Matérn Gaussian random fields, where A and à are elliptic second-order differential operators, formulated on a bounded Euclidean domain D ⊂ Rd and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle–Matérn fields.