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ESSEC Working Papers,
ESSEC Research Center, ESSEC Business School

No WP1416: On the Capacity Functional of Excursion Sets of Gaussian Random Fields on R²

Marie Kratz () and Werner Nagel ()
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Marie Kratz: ESSEC Business School, Postal: AVENUE BERNARD HIRSCH, CS 50105 CERGY, 95021 CERGY PONTOISE CEDEX, FRANCE
Werner Nagel: Institut fur Stochastik Ernst Friedrich-Schiller-Universität Jena, Postal: Ernst-Abbe-Platz 2, 07743 Jena, DEUTSCHLAND

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.

Keywords: Capacity functional; Crossings; Excursion set; Gaussian field; Growing circle method; Rice formulas; Second moment measure; Sweeping line method; Stereology; Stochastic geometry

JEL-codes: C12; C32

19 pages, November 2014

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