Random measurable sets and covariogram realizability problems. (September 2015)

• Record Type: Journal Article
• Authors: Galerne, Bruno; Lachièze-Rey, Raphael
• Is Part Of: Advances in applied probability. Volume 47:Number 3(2015)
• Page Start: 611
• Page End: 639
• Publication Date: 2015-09
• Subjects: Random measurable set, -- realizability, -- S2 problem, -- covariogram, -- perimeter, -- truncated moment problem; 60D05, -- 28C05; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1239/aap/1444308874
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract : We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to the S 2 problem in materials science. Our method is based on the covariogram functional for random measurable sets (RAMS) and on a result about the representation of positive operators on a noncompact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, and they provide a weaker framework that allows the manipulation of more irregular functionals, such as the perimeter. We therefore use the illustration provided by the S 2 problem to advocate the use of RAMS for solving theoretical problems of a geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.