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Limit Theorems for Excursion Sets of Stationary Random Fields

In: Modern Stochastics and Applications

Author

Listed:
  • Evgeny Spodarev

    (Ulm University)

Abstract

We give an overview of the recent asymptotic results on the geometry of excursion sets of stationary random fields. Namely, we cover a number of limit theorems of central type for the volume of excursions of stationary (quasi-, positively or negatively) associated random fields with stochastically continuous realizations for a fixed excursion level. This class includes in particular Gaussian, Poisson shot noise, certain infinitely divisible, α-stable, and max-stable random fields satisfying some extra dependence conditions. Functional limit theorems (with the excursion level being an argument of the limiting Gaussian process) are reviewed as well. For stationary isotropic C 1-smooth Gaussian random fields similar results are available also for the surface area of the excursion set. Statistical tests of Gaussianity of a random field which are of importance to real data analysis as well as results for an increasing excursion level round up the paper.

Suggested Citation

  • Evgeny Spodarev, 2014. "Limit Theorems for Excursion Sets of Stationary Random Fields," Springer Optimization and Its Applications, in: Volodymyr Korolyuk & Nikolaos Limnios & Yuliya Mishura & Lyudmyla Sakhno & Georgiy Shevchenko (ed.), Modern Stochastics and Applications, edition 127, pages 221-241, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-03512-3_13
    DOI: 10.1007/978-3-319-03512-3_13
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    Cited by:

    1. Koch, Erwan & Dombry, Clément & Robert, Christian Y., 2019. "A central limit theorem for functions of stationary max-stable random fields on Rd," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3406-3430.
    2. Andriy Olenko & Dareen Omari, 2020. "Reduction Principle for Functionals of Vector Random Fields," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 573-598, June.

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