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Nonstationarity in ℝn is second-order stationarity in ℝ2n

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Olivier Perrin  and
Wendy Meiring
Olivier Perrin*
Affiliation:
Université Toulouse 1
Wendy Meiring*
Affiliation:
University of California, Santa Barbara
*
∗Postal address: GREMAQ, Université Toulouse 1, 21, Allée de Brienne, 31000 Toulouse, France. Email address: perrin@cict.fr
∗∗Postal address: Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA
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Abstract

We prove that a nonstationary random field indexed by ℝn with moments at least of order 2 can always be viewed as second-order stationary in ℝ2n.

Keywords

Nonstationaritybijective transformationsecond-order stationarity

MSC classification

Primary: 60G60: Random fields
Secondary: 60G12: General second-order processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 815 - 820
Copyright
Copyright © Applied Probability Trust 2003 

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References

Cressie, N. A. C. (1993). Statistics for Spatial Data, revised edn. John Wiley, New York.Google Scholar
Dickinson, B. W. (1980). Two-dimensional Markov spectrum estimates need not exist. IEEE Trans. Inf. Theory 26, 120–121.Google Scholar
Gneiting, T. (2002). Compactly supported correlation functions. J. Multivariate Anal. 83, 493–508.Google Scholar
Gneiting, T. (2002). Nonseparable, stationary covariance functions for space-time data. J. Amer. Statist. Assoc. 97, 590–600.Google Scholar
Gneiting, T. and Sasvári, Z. (1999). The characterization problem for isotropic covariance functions. Math. Geology 31, 105–111.Google Scholar
Krein, M. (1940). Sur le problème du prolongement des fonctions hermitiennes positives et continues. C. R. (Doklady) Acad. Sci. URSS 26, 17–22.Google Scholar
Rudin, W. (1963). The extension problem for positive-definite functions. Illinois J. Math. 7, 532–539.CrossRefGoogle Scholar
Rudin, W. (1970). An extension theorem for positive-definite functions. Duke Math. J. 37, 49–53.CrossRefGoogle Scholar
Sampson, P. D., and Guttorp, P. (1992). Nonparametric estimation of non-stationary spatial covariance structure. J. Amer. Statist. Assoc. 87, 108–119.Google Scholar
Sampson, P. D., Damian, D., and Guttorp, P. (2001). Advances in modeling and inference for environmental processes with nonstationary spatial covariance. In GeoENV III-Geostatistics for Environmental Applications, eds Monestiez, P., Allard, D. and Froidevaux, R., Kluwer, Dordrecht, pp. 17–32.CrossRefGoogle Scholar
Sasvári, Z. (1986). The extension problem for measurable positive definite functions. Math. Z. 191, 475–478.Google Scholar
Sasvári, Z. (1994). Positive Definite and Definitizable Functions (Math. Topics 2). Akademie, Berlin.Google Scholar
Seghier, A. (1991). Extension de fonctions de type positif et entropie associée. Cas multidimensionnel. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 651–675.CrossRefGoogle Scholar