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Equivalence of Gaussian measures for some nonstationary random fields
Institution:1. University of Oviedo, Department of Statistics and Operations Research, C-Calvo Sotelo, s/n, 33007 Oviedo, Spain;2. Istituto Dalle Molle di Studi sull''Intelligenza Artificiale (IDSIA), Galleria 2, 6928 Manno (Lugano), Switzerland;1. Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK, USA;2. Department of Mathematics, Washington State University, Pullman, WA, USA;1. Loughborough University, United Kingdom;2. University of Sheffield, United Kingdom
Abstract:Gaussian random fields whose covariance structures are described by a power law model provide a simple and flexible class of models for isotropic random fields. This class includes fractional Brownian fields as a special case. Because these random fields are nonstationary, the extensive results available on equivalence of Gaussian measures for stationary models do not apply to them. This work shows that results on equivalence for two stationary Gaussian random field models extend in a natural way to the equivalence of a stationary model and a power law model. This result is used to show that if we use a power law model for predicting a random field at unobserved locations when in fact the random field is stationary, we can obtain asymptotically optimal predictions as long as the high frequency behavior of the true spectral density is sufficiently close to the high frequency behavior of the spectral density of the power law model.
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