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February 1999 Predicting random fields with increasing dense observations
Michael L. Stein
Ann. Appl. Probab. 9(1): 242-273 (February 1999). DOI: 10.1214/aoap/1029962604


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.


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Michael L. Stein. "Predicting random fields with increasing dense observations." Ann. Appl. Probab. 9 (1) 242 - 273, February 1999.


Published: February 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0955.62095
MathSciNet: MR1682572
Digital Object Identifier: 10.1214/aoap/1029962604

Primary: 62M20
Secondary: 41A25 , 62M40

Keywords: Approximation in Hilbert spaces , design of time series experiments , fixed-domain asymptotics , infill asymptotics , kriging , sampling theorem

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.9 • No. 1 • February 1999
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