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On Extreme Values in Stationary Random Fields

  • M. R. Leadbetter
  • Holger Rootzén
Part of the Trends in Mathematics book series (TM)

Summary

This paper develops distributional extremal theory for maxima M T = max(X t: 0 ⩽ t ⩽ T) of a stationary random field X t. A general form of “extremal types theorem” is proven and shown to apply to M T under very weak dependence restrictions. That is, any non-degenerate distributional limit for the normalized family a T(MT - b T) (a T > 0) must be one of the three classical types. Domain of attraction criteria are discussed.

The dependence structure used here for fields involves a potentially very weak type of strong-mixing, “Coordinatewise (Cw) mixing”) using mild individual “past-future” conditions in each coordinate direction. Together with careful control of numbers and sizes of sets involved, this avoids the over-restrictive nature of common generalizations of mixing conditions to apply to random fields. Futher, the conditions may be readily adpated to deal with other quite general problems of Centeral Limit type (cf. [6]).

Keywords

Random Field Local Dependence Extremal Index Extremal Type Asymptotic Independence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    Adler, R.A., The geometry of random fields, John Wiley, New York, 1981.zbMATHGoogle Scholar
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    Bolthausen E., On the central limit theorem for stationary mixing random fields, Ann. Prob. 10, (1982) 1047–1050.MathSciNetzbMATHCrossRefGoogle Scholar
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    Doukhan, P., Mixing: Properties and Examples, Springer Lecture Notes in Statistics #85, 1995.Google Scholar
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    Guyon, X., Random Fields on a Network: Modeling, Statistics and Applications, Springer-Verlag, 1995.Google Scholar
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    Leadbetter M.R., Lindgren G., Rootzen H., Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, New York, 1983.zbMATHCrossRefGoogle Scholar
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    Leadbetter MR, Rootzén H, Choi H, Coordinatewise mixing and Central Limit Theory for additive random set functions on R d, in preparationGoogle Scholar
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    Piterbarg, V.I., Asymptotic methods in the theory of Gaussian processes and fields, Trans. of Math. Monographs 148, American Mathematical Society, 1996.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • M. R. Leadbetter
    • 1
  • Holger Rootzén
    • 2
  1. 1.Department of StatisticsUniversity of North Carolina at Chapel HillChapel HillUSA
  2. 2.Mathematics DepartmentChalmers University of TechnologyGothenburgSweden

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