Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces

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.