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 is an element of X )indexed by a compact metric space (X, d(X)) can be obtained if the mean value function m : chi -> R and the covariance function Q: X x X -> R of Z are known. We consider the problem of predicting the value of Z (x*) at some location x* is an element of X based on observations at locations {x(j)}(j=1)(n), which accumulate at x* as n -> infinity (or, more generally, predicting phi(Z) based on {phi(j)(Z)}(j=i)(n) for linear functionals phi, phi(1), ..., phi(n)). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure ((m) over tilde, (rho) over tilde), without any restrictive assumptions on rho, (rho) over tilde such as stationarity. We, for the first time, provide necessary and sufficient conditions on ((m) over tilde, (rho) over tilde) for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to phi. These general results are illustrated by weakly stationary random fields on X subset of R-d with Matern or periodic covariance functions, and on the sphere X = S-2 for the case of two isotropic covariance functions