Asymptotic Properties of Linear Predictors

Suppose we observe a Gaussian random field Z with mean function m and covariance function K at some set of locations. Call the pair (m, K) the second-order structure of the random field. If (m, K) is known, then as noted in 1.2, the prediction of Z at unobserved locations is just a matter of calculation. To review, the conditional distribution of Z at an unobserved location is normal with conditional mean that is a linear function of the observations and constant conditional variance. In practice, (m, K) is at least partially unknown and it is usually necessary to estimate (m, K) from the same data we use to do the prediction. Thus, it might be natural to proceed immediately to methods for estimating second-order structures of Gaussian random fields. However, until we know something about the relationship between the second-order structure and linear predictors, it will be difficult to judge what is meant by a good estimate of the second-order structure. In particular, it will turn out that it is possible to get (m, K) nonnegligibly wrong and yet still get nearly optimal linear predictors. More specifically, for a random field possessing an autocovariance function, if the observations are tightly packed in a region in which we wish to predict the random field, then the low frequency behavior of the spectrum has little impact on the behavior of the optimal linear predictions.