, Volume 20, Issue 2, pp 333–392 | Cite as

Extremes of Gaussian random fields with regularly varying dependence structure

  • Krzysztof Dȩbicki
  • Enkelejd Hashorva
  • Peng Liu


Let \(X(t), t\in \mathcal {T}\) be a centered Gaussian random field with variance function σ 2(⋅) that attains its maximum at the unique point \(t_{0}\in \mathcal {T}\), and let \(M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)\). For \(\mathcal {T}\) a compact subset of ℝ, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 − σ(t) has a polynomial decrease to 0 as tt 0. In this contribution we consider more general case that 1 − σ(t) is regularly varying at t 0. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), deriving the exact tail asymptotics of \(M(\mathcal {T})\) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t 0. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.


Non-stationary Gaussian processes Gaussian random fields Extremes Fractional Brownian motion Regular variation Uniform approximation 

AMS 2000 Subject Classifications

Primary 60G15 secondary 60G70 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Krzysztof Dȩbicki
    • 1
  • Enkelejd Hashorva
    • 2
  • Peng Liu
    • 1
    • 2
  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Department of Actuarial ScienceUniversity of LausanneUNIL-DorignySwitzerland

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