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]).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adler, R.A., The geometry of random fields, John Wiley, New York, 1981.
Bolthausen E., On the central limit theorem for stationary mixing random fields, Ann. Prob. 10, (1982) 1047–1050.
Doukhan, P., Mixing: Properties and Examples, Springer Lecture Notes in Statistics #85, 1995.
Guyon, X., Random Fields on a Network: Modeling, Statistics and Applications, Springer-Verlag, 1995.
Leadbetter M.R., Lindgren G., Rootzen H., Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, New York, 1983.
Leadbetter MR, Rootzén H, Choi H, Coordinatewise mixing and Central Limit Theory for additive random set functions on R d, in preparation
Piterbarg, V.I., Asymptotic methods in the theory of Gaussian processes and fields, Trans. of Math. Monographs 148, American Mathematical Society, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Leadbetter, M.R., Rootzén, H. (1998). On Extreme Values in Stationary Random Fields. In: Karatzas, I., Rajput, B.S., Taqqu, M.S. (eds) Stochastic Processes and Related Topics. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2030-5_15
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2030-5_15
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7389-9
Online ISBN: 978-1-4612-2030-5
eBook Packages: Springer Book Archive