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A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail

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  • Azaïs, Jean-Marc
  • Wschebor, Mario

Abstract

We study the probability distribution F(u) of the maximum of smooth Gaussian fields defined on compact subsets of having some geometric regularity. Our main result is a general expression for the density of F. Even though this is an implicit formula, one can deduce from it explicit bounds for the density, and hence for the distribution, as well as improved expansions for 1-F(u) for large values of u. The main tool is the Rice formula for the moments of the number of roots of a random system of equations over the reals. This method enables also to study second-order properties of the expected Euler characteristic approximation using only elementary arguments and to extend these kinds of results to some interesting classes of Gaussian fields. We obtain more precise results for the "direct method" to compute the distribution of the maximum, using the spectral theory of GOE random matrices.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:7:p:1190-1218
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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.
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    Cited by:

    1. Lachièze-Rey, Raphaël, 2019. "Bicovariograms and Euler characteristic of random fields excursions," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4687-4703.
    2. Cheng, Dan, 2016. "Excursion probability of certain non-centered smooth Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 883-905.
    3. Wenbo V. Li & Ang Wei, 2012. "A Gaussian Inequality for Expected Absolute Products," Journal of Theoretical Probability, Springer, vol. 25(1), pages 92-99, March.
    4. Azaïs, Jean-Marc & Delmas, Céline, 2022. "Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 411-445.
    5. 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.

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    1. 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.

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