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

We consider two Gaussian measures mu, (mu) over tilde on a separable Hilbert space, with fractional-order covariance operators A(-2 beta) and (A) over tilde (-2 beta), respectively, and derive necessary and sufficient conditions on A, (A) over tilde and beta, (beta) over tilde > 0 for I. euivalence of the measures mu and (mu) over tilde, and II. uniform asymptotic optimality of linear predictions for mu based on the misspecified measure (mu) over tilde. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle-Matern Gaussian random fields, where A and (A) over tilde are elliptic second-order differential operators, formulated on a bounded Euclidean domain D subset of 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-Matern fields.

Automated summary

We consider two Gaussian measures mu and (mu) over tilde on a separable Hilbert space, with fractional-order covariance operators A(-2 beta) and (A) over tilde (-2 beta), respectively, and derive necessary and sufficient conditions for the equivalence of the measures and the uniform asymptotic optimality of linear predictions. These results hold for Gaussian processes on compact metric spaces, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle-Matern fields.