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

Optimal linear prediction (also known as kriging) of a random field ${Z(x)}_{xinmathcal{X}}$ indexed by a compact metric space $(mathcal{X},d_{mathcal{X}})$ can be obtained if the mean value function $mcolonmathcal{X}tomathbb{R}$ and the covariance function $varrhocolonmathcal{X}timesmathcal{X}tomathbb{R}$ of $Z$ are known. We consider the problem of predicting the value of $Z(x^*)$ at some location $x^*inmathcal{X}$ based on observations at locations ${x_j}_{j=1}^n$ which accumulate at $x^*$ as $ntoinfty$ (or, more generally, predicting $varphi(Z)$ based on ${varphi_j(Z)}_{j=1}^n$ for linear functionals $varphi, varphi_1, ldots, varphi_n$). Our main result characterizes the asymptotic performance of linear predictors (as $n$ increases) based on an incorrect second order structure $(tilde{m},tilde{varrho})$, without any restrictive assumptions on $varrho, tilde{varrho}$ such as stationarity. We, for the first time, provide necessary and sufficient conditions on $(tilde{m},tilde{varrho})$ for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to $varphi$. These general results are illustrated by weakly stationary random fields on $mathcal{X}subsetmathbb{R}^d$ with Matern or periodic covariance functions, and on the sphere $mathcal{X}=mathbb{S}^2$ for the case of two isotropic covariance functions.