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Asymptotic formula for the tail of the maximum of smooth stationary Gaussian fields on non locally convex sets

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  • Azaïs, Jean-Marc
  • Pham, Viet-Hung

Abstract

In this paper we consider the distribution of the maximum of a Gaussian field defined on non locally convex sets. Adler and Taylor or Azaïs and Wschebor give the expansions in the locally convex case. The present paper generalizes their results to the non locally convex case by giving a full expansion in dimension 2 and some generalizations in higher dimension. For a given class of sets, a Steiner formula is established and the correspondence between this formula and the tail of the maximum is proved. The main tool is a recent result of Azaïs and Wschebor that shows that under some conditions the excursion set is close to a ball with a random radius. Examples are given in dimension 2 and higher.

Suggested Citation

  • Azaïs, Jean-Marc & Pham, Viet-Hung, 2016. "Asymptotic formula for the tail of the maximum of smooth stationary Gaussian fields on non locally convex sets," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1385-1411.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:5:p:1385-1411
    DOI: 10.1016/j.spa.2015.11.007
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    References listed on IDEAS

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    1. Rychlik, Igor, 1990. "New bounds for the first passage, wave-length and amplitude densities," Stochastic Processes and their Applications, Elsevier, vol. 34(2), pages 313-339, April.
    2. Azaïs, Jean-Marc & Wschebor, Mario, 2008. "A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1190-1218, July.
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    Cited by:

    1. Elena Di Bernardino & Céline Duval, 2022. "Statistics for Gaussian random fields with unknown location and scale using Lipschitz‐Killing curvatures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 143-184, March.

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