, Volume 20, Issue 4, pp 807–838 | Cite as

Clustering of high values in random fields

  • Luísa Pereira
  • Ana Paula Martins
  • Helena Ferreira


The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with \(\mathbb {Z}^{2}\), and that they satisfy stationarity and isotropy conditions. Here we extend the existing theory, concerning the asymptotic behavior of the maximum and the extremal index, to non-stationary and anisotropic random fields, defined over discrete subsets of \(\mathbb {R}^{2}\). We show that, under a suitable coordinatewise mixing condition, the maximum may be regarded as the maximum of an approximately independent sequence of submaxima, although there may be high local dependence leading to clustering of high values. Under restrictions on the local path behavior of high values, criteria are given for the existence and value of the spatial extremal index which plays a key role in determining the cluster sizes and quantifying the strength of dependence between exceedances of high levels. The general theory is applied to the class of max-stable random fields, for which the extremal index is obtained as a function of well-known tail dependence measures found in the literature, leading to a simple estimation method for this parameter. The results are illustrated with non-stationary Gaussian and 1-dependent random fields. For the latter, a simulation and estimation study is performed.


Random field Max-stable process Extremal dependence Spatial extremal index 

AMS 2000 Subject Classifications

60G60 60G70 


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We are grateful to the referees for their detailed comments and important suggestions which significantly improved this paper. This research was supported by National Foundation of Science and Technology through UID/MAT/00212/2013.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Luísa Pereira
    • 1
  • Ana Paula Martins
    • 1
  • Helena Ferreira
    • 1
  1. 1.Universidade da Beira InteriorCentro de Matemática e Aplicações (CMA-UBI)CovilhãPortugal

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