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On the Maximum and Minimum of a Stationary Random Field

  • Luísa Pereira
Conference paper
Part of the Studies in Theoretical and Applied Statistics book series (STAS)

Abstract

We determine the class of nondegenerate joint-limiting distributions for the maximum and minimum of stationary random fields \(\mathbf{X} = \left \{X_{\mathbf{n}}\right \}_{\mathbf{n\in {\mathbb{N}}^{2}}}\) satisfying a long-range dependence restriction for each coordinate direction at a time. Unlike the classical result for i.i.d. random fields the maximum and minimum of X may be asymptotically dependent. We also give a sufficient condition for the asymptotic independence of the maximum and minimum. Additional conditions are given in order to obtain the asymptotic independence of the locations of maximum and minimum.

Notes

Acknowledgements

We would like to thank the referee’s careful reading of the manuscript which has resulted in improvements to the final form of this chapter.

This research was supported by the research unit “Centro de Matemática” of the University of Beira Interior through the Foundation for Science and Technology (FCT).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.University of Beira InteriorCovilhãPortugal

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