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 ⊂ 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.

Optimal linear prediction (aka. kriging) of a random field {Z(x)} x∈X indexed by a compact metric space (X, dX ) can be obtained if the mean value function m: X →R and the covariance function ∂: X × X →R of Z are known. We consider the problem of predicting the value of Z(x*) at some location x*∈ X based on observations at locations {xj }nj =1, which accumulate at x*as n→∞(or, more generally, predicting φ(Z) based on {φj (Z)}nj =1 for linear functionals φ,φ1, . . . , φn). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure (m, ∂), without any restrictive assumptions on ,∂ ∂ such as stationarity.We, for the first time, provide necessary and sufficient conditions on (m,∂) for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to φ. These general results are illustrated by weakly stationary random fields on X ⊂ Rd with Matérn or periodic covariance functions, and on the sphere X = S2 for the case of two isotropic covariance functions.