Rataj, Jan Mean Euler characteristic of stationary random closed sets. (English) Zbl 1469.60059 Stochastic Processes Appl. 137, 252-271 (2021). Summary: The translative intersection formula of integral geometry yields an expression for the mean Euler characteristic of a stationary random closed set intersected with a fixed observation window. We formulate this result in the setting of sets with positive reach and using flag measures which yield curvature measures as marginals. As an application, we consider excursion sets of stationary random fields with \(C^{1,1}\) realizations, in particular, stationary Gaussian fields, and obtain results which extend those known from the literature. MSC: 60D05 Geometric probability and stochastic geometry 60G60 Random fields 53C65 Integral geometry Keywords:set with positive reach; curvature measure; flag measure; Euler characteristic; random closed set; Gaussian random field PDFBibTeX XMLCite \textit{J. Rataj}, Stochastic Processes Appl. 137, 252--271 (2021; Zbl 1469.60059) Full Text: DOI arXiv References: [1] Adler, R. J., The Geometry of Random Fields. Wiley Series in Probability and Mathematical Statistics (1981), J. Wiley & Sons, Ltd.: J. Wiley & Sons, Ltd. Chichester [2] Adler, R. J.; Taylor, J. E., Random Fields and Geometry (2007), Springer · Zbl 1149.60003 [3] Cambanis, S., On some continuity and differentiability properties of paths of Gaussian processes, J. Multivariate Anal., 3, 420-434 (1973) · Zbl 0272.60035 [4] Federer, H., Curvature measures, Trans. Amer. Math. Soc., 93, 418-491 (1959) · Zbl 0089.38402 [5] Federer, H., Geometric Measure Theory (1969), Springer: Springer Berlin · Zbl 0176.00801 [6] Goodey, P.; Hinderer, W.; Hug, D.; Rataj, J.; Weil, W., A flag representation of projection functions, Adv. Geom., 17, 303-322 (2017) · Zbl 1401.52008 [7] Hug, D.; Rataj, J.; Weil, W., A product integral representation of mixed volumes of two convex bodies, Adv. Geom., 13, 633-662 (2013) · Zbl 1284.52006 [8] Lachièze-Rey, R., Bicovariograms and Euler characteristic of random fields excursions, Stochastic Process. Appl., 129, 4687-4703 (2019) · Zbl 1448.60112 [9] Rataj, J.; Zähle, M., Mixed curvature measures for sets of positive reach and a translative integral formula, Geom. Dedicata, 57, 259-283 (1995) · Zbl 0844.53050 [10] Rataj, J.; Zähle, M., Curvature Measures of Singular Sets (2019), Springer · Zbl 1423.28001 [11] Schneider, R., Convex bodies: The Brunn-Minkowski theory, (Encyclopedia of Mathematics and Its Applications, Vol. 151 (2014), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1287.52001 [12] Schneider, R.; Weil, W., Stochastic and Integral Geometry (2008), Springer: Springer New York · Zbl 1175.60003 [13] Zähle, M., Curvature measures and random sets. II, Probab. Theory Relat. Fields, 71, 37-58 (1986) · Zbl 0554.60017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.