Asymptotically Efficient Prediction of a Random Field with a Misspecified Covariance Function

Best linear unbiased predictors of a random field can be obtained if the covariance function of the random field is specified correctly. Consider a random field defined on a bounded region R. We wish to predict the random field z(·) at a point x in R based on observations z(x₁), z(x₂), …, z(xN) in R, where ᵢ\^∞i = 1 has x as a limit point but does not contain x. Suppose the covariance function is misspecified, but has an equivalent (mutually absolutely continuous) corresponding Gaussian measure to the true covariance function. Then the predictor of $z(x)$ based on z(x₁), …, z(xN) will be asymptotically efficient as N tends to infinity.