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Limit Theorems for Excursion Sets of Stationary Random Fields

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Modern Stochastics and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 90))

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.

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Author information

Authors and Affiliations

  1. Institute of Stochastics, Ulm University, 89069, Ulm, Germany

    Evgeny Spodarev

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  1. Evgeny Spodarev
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Correspondence to Evgeny Spodarev .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Volodymyr Korolyuk

  2. Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, Compiegne Cedex, France

    Nikolaos Limnios

  3. Dept. of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Yuliya Mishura

  4. Dept.of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Lyudmyla Sakhno

  5. Dept.of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Georgiy Shevchenko

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Spodarev, E. (2014). Limit Theorems for Excursion Sets of Stationary Random Fields. In: Korolyuk, V., Limnios, N., Mishura, Y., Sakhno, L., Shevchenko, G. (eds) Modern Stochastics and Applications. Springer Optimization and Its Applications, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-03512-3_13

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