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Bicovariograms and Euler characteristic of random fields excursions

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  • Lachièze-Rey, Raphaël

Abstract

Let f be a C1 bivariate function with Lipschitz derivatives, and F={x∈R2:f(x)⩾λ} an upper level set of f, with λ∈R. We present a new identity giving the Euler characteristic of F in terms of its three-points indicator functions. A bound on the number of connected components of F in terms of the values of f and its gradient, valid in higher dimensions, is also derived. In dimension 2, if f is a random field, this bound allows to pass the former identity to expectations if f’s partial derivatives have Lipschitz constants with finite moments of sufficiently high order, without requiring bounded conditional densities. This approach provides an expression of the mean Euler characteristic in terms of the field’s third order marginal. Sufficient conditions and explicit formulas are given for Gaussian fields, relaxing the usual C2 Morse hypothesis.

Suggested Citation

  • Lachièze-Rey, Raphaël, 2019. "Bicovariograms and Euler characteristic of random fields excursions," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4687-4703.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:11:p:4687-4703
    DOI: 10.1016/j.spa.2018.12.006
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    References listed on IDEAS

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    1. 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.
    2. C. H. Arns & J. Mecke & K. Mecke & D. Stoyan, 2005. "Second-order analysis by variograms for curvature measures of two-phase structures," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 47(3), pages 397-409, October.
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    More about this item

    Keywords

    Random fields; Euler characteristic; Gaussian processes; Covariograms; Intrinsic volumes; C1; 1 functions;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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