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 $\{Z(x)\}_{x\in\mathcal{X}}$ indexed by a compact metric space $(\mathcal{X},d_{\mathcal{X}})$ can be obtained if the mean value function $m\colon\mathcal{X}\to\mathbb{R}$ and the covariance function $\varrho\colon\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ of $Z$ are known. We consider the problem of predicting the value of $Z(x^*)$ at some location $x^*\in\mathcal{X}$ based on observations at locations $\{x_j\}_{j=1}^n$ which accumulate at $x^*$ as $n\to\infty$ (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 $\mathcal{X}\subset\mathbb{R}^d$ with Mat\'ern or periodic covariance functions, and on the sphere $\mathcal{X}=\mathbb{S}^2$ for the case of two isotropic covariance functions.