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

Abstract

We consider Gaussian measures $\mu, \tilde{\mu}$ on a separable Hilbert space, with fractional-order covariance operators $A^{-2\beta}$ resp. $\tilde{A}^{-2\tilde{\beta}}$, and derive necessary and sufficient conditions on $A, \tilde{A}$ and $\beta, \tilde{\beta} > 0$ for I. equivalence of the measures $\mu$ and $\tilde{\mu}$, and II. uniform asymptotic optimality of linear predictions for $\mu$ based on the misspecified measure $\tilde{\mu}$. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle-Mat\'ern 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-Mat\'ern fields.