Predicting random fields with increasing dense observations

Abstract

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