Convergence of the Best Linear Predictor of a Weakly Stationary Random Field

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Abstract

Suppose that \(\{X_{m,n}\}_{(m,n)\in \mathbb {Z}^2}\) is a centered, weakly stationary random field with spectral density function W. Let \(X'\) denote the best linear estimate of \(X_{0,0}\) based on those \(X_{m,n}\) where the pair (mn) varies over a lexicographical halfplane of \(\mathbb {Z}^2\). The convergence rate of the partial sums for \(X'\) is studied, in relation to the analytical properties of W. Under the assumption that W is bounded away from zero and infinity, it is shown that the faster the convergence rate, the smoother W is. To be more precise, the partial sums of \(X'\) converge at an exponential rate if and only if W has an analytic extension to a neighborhood of the torus \(\mathbb {T}^2\); the partial sums of \(X'\) converge at a polynomial rate if and only if W satisfies a generalized Hölder condition. The convergence rate of \(X'\) is also characterized in terms of the related function \(\Phi \), where \(W = |\Phi |^2\) on \(\mathbb {T}^2\), the Fourier coefficients of \(\Phi \) vanish outside the halfplane, and \(\Phi \) has a certain optimality property.

Keywords

Hölder class Halfplane Prediction Random field Weakly stationary 

Mathematics Subject Classification

42B35 60G25 60G60 

Notes

Acknowledgements

The author expresses his heartfelt thanks the Referees for their numerous helpful comments and suggestions.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsOld Dominion UniversityNorfolkUSA

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