Predicting random fields with increasing dense observations

This work investigates some spectral characteristics of the errors of optim al linear predictors for weakly stationary random fields. More specifically , for errors of optimal linear predictors, results here explicitly bound th e fraction of the variance attributable to some set of frequencies. Such a bound is first obtained for random fields on R-d observed on the infinite l attice delta J for all J on the d-dimensional integer lattice. If the spect ral density exists, then the faster the spectral density tends to 0 at high frequencies, the more quickly this bound tends to 0 as delta down arrow 0. Under certain conditions on the spectral density, a similar result is give n for processes on R where both observations and predictands are confined t o a finite interval and observations may not be evenly spaced. These result s provide a powerful tool for studying a problem the author has previously addressed using different methods: the properties of linear predictors calc ulated under an incorrect spectral density. Specifically, this work gives a number of new rates of convergence to optimality for predictors based on a n incorrect spectral density when the ratio of the incorrect to the correct spectral density tends to 1 at high frequencies.