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

Abstract

We consider two Gaussian measures $\mathrm{\mu },\tilde{\mathrm{\mu }}$ on a separable Hilbert space, with fractional-order covariance operators $A^{-2\mathit{\beta }}$ and ${\tilde{A}}^{-2\tilde{\mathit{\beta }}}$, respectively, and derive necessary and sufficient conditions on $A,\tilde{A}$ and $\mathit{\beta },\tilde{\mathit{\beta }}>0$ for I. equivalence of the measures μ and $\tilde{\mathrm{\mu }}$, and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure $\tilde{\mathrm{\mu }}$. 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 $\tilde{A}$ are elliptic second-order differential operators, formulated on a bounded Euclidean domain $\mathcal{D}\subset {\mathbb{R}}^d$ 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.