High level excursion set geometry for non-Gaussian infinitely divisible random fields

• Autores: Robert J. Adler, Gennady Samorodnitsky, Jonathan E. Taylor
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 1, 2013, págs. 134-169
• Idioma: inglés
• DOI: 10.1214/11-AOP738
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• Resumen

  • We consider smooth, infinitely divisible random fields (X(t),t∈M), M⊂Rd, with regularly varying Lévy measure, and are interested in the geometric characteristics of the excursion sets Au={t∈M:X;(t)>u} over high levels u.

    For a large class of such random fields, we compute the u→∞ asymptotic joint distribution of the numbers of critical points, of various types, of X in Au, conditional on Au being nonempty. This allows us, for example, to obtain the asymptotic conditional distribution of the Euler characteristic of the excursion set.

    In a significant departure from the Gaussian situation, the high level excursion sets for these random fields can have quite a complicated geometry. Whereas in the Gaussian case nonempty excursion sets are, with high probability, roughly ellipsoidal, in the more general infinitely divisible setting almost any shape is possible.