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On the Capacity Functional of Excursion Sets of Gaussian Random Fields on R²

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Abstract

When a random field (Xt; t 2 R2) is thresholded on a given level u, the excursion set is given by its indicator 1[u;1)(Xt). The purpose of this work is to study functionals (as established in stochastic geometry) of these random excursion sets, as e.g. the capacity functional as well as the second moment measure of the boundary length. It extends results obtained for the one-dimensional case to the two-dimensional case, with tools borrowed from crossings theory, in particular Rice methods, and from integral and stochastic geometry.

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  • Kratz, Marie & Nagel , Werner, 2014. "On the Capacity Functional of Excursion Sets of Gaussian Random Fields on R²," ESSEC Working Papers WP1416, ESSEC Research Center, ESSEC Business School.
    • Handle: RePEc:ebg:essewp:dr-14016
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    File URL: https://hal-essec.archives-ouvertes.fr/hal-01085072/document
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    Keywords

    Capacity functional; Crossings; Excursion set; Gaussian field; Growing circle method; Rice formulas; Second moment measure; Sweeping line method; Stereology; Stochastic geometry;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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