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Random measurable sets and covariogram realizability problems

Part of: Geometric probability and stochastic geometry Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  21 March 2016

Bruno Galerne  and
Raphael Lachièze-Rey
Bruno Galerne*
Affiliation:
Université Paris Descartes
Raphael Lachièze-Rey*
Affiliation:
Université Paris Descartes
*
∗ Postal address: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, 45 Rue des Saints-pères, 75006 Paris, France.
∗ Postal address: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, 45 Rue des Saints-pères, 75006 Paris, France.
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Abstract

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We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to the S2 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 S2 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.

Keywords

Random measurable setrealizabilityS2 problemcovariogramperimetertruncated moment problem

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 28C05: Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 611 - 639
Copyright
Copyright © Applied Probability Trust 2015 

References

Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edn. Springer, Berlin.Google Scholar
Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press.Google Scholar
Bogachev, V. I. (2007). Measure Theory, Vol. II. Springer, Berlin.Google Scholar
Caselles, V., Chambolle, A., Moll, S. and Novaga, M. (2008). A characterization of convex calibrable sets in R N with respect to anisotropic norms. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 803-832.Google Scholar
Chilès, J.-P. and Delfiner, P. (1999). Geostatistics: Modeling Spatial Uncertainty. John Wiley, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II: General Theory and Structure, 2nd edn. Springer, New York.Google Scholar
Deza, M. M. and Laurent, M. (1997). Geometry of Cuts and Metrics. Springer, Berlin.Google Scholar
Emery, X. (2010). On the existence of mosaic and indicator random fields with spherical, circular, and triangular variograms. Math. Geosci. 42, 969–984.Google Scholar
Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. CRC, Boca Raton, FL.Google Scholar
Fritz, T. and Chaves, R. (2013). Entropic inequalities and marginal problems. IEEE Trans. Inf. Theory 59, 803–817.Google Scholar
Galerne, B. (2011). Computation of the perimeter of measurable sets via their covariogram. Applications to random sets. Image Anal. Stereol. 30, 39–51.Google Scholar
Galerne, B. (2014). Random fields of bounded variation and computation of their variation intensity. Tech. Rep. 2014-25, Laboratoire MAP5, Université Paris Descartes.Google Scholar
Himmelberg, C. J. (1975). Measurable relations. Fund. Math 87, 53–72.Google Scholar
Hirsch, F. and Lacombe, G. (1999). Elements of Functional Analysis (Graduate Texts Math. 192), Springer, New York.Google Scholar
Jiao, Y., Stillinger, F. H. and Torquato, S. (2007). Modeling heterogeneous materials via two-point correlation functions: basic principles. Phys. Rev. E 76, 031110.Google Scholar
Kallenberg, O. (1986). Random Measures, 4th edn. Akademie-Verlag, Berlin.Google Scholar
Kuna, T., Lebowitz, J. L. and Speer, E. R. (2011). Necessary and sufficient conditions for realizability of point processes. Ann. Appl. Prob. 21, 1253–1281.CrossRefGoogle Scholar
Lachièze-Rey, R. (2013). Realisability conditions for second-order marginals of biphased media. Random Structures Algorithms 10.1002/rsa.20546.Google Scholar
Lachiéze-Rey, R. and Molchanov, I. (2015). Regularity conditions in the realisability problem with applications to point processes and random closed sets. Ann. Appl. Prob. 25, 116–149.Google Scholar
Lantuéjoul, C. (2002). Geostatistical Simulation: Models and Algorithms. Springer, Berlin.Google Scholar
Masry, E. (1972). On covariance functions of unit processes. SIAM J. Appl. Math. 23, 28–33.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Matheron, G. (1993). Une conjecture sur la covariance d'un ensemble aléatoire. In Cahiers de Géostatistique, Fascicule 3, Compte-Rendu des Journées de Géostatistique (Fontainebleau, 1993), pp. 107–113.Google Scholar
McMillan, B. (1955). History of a problem. J. Soc. Ind. Appl. Math. 3, 119–128.Google Scholar
Molchanov, I. (2005). Theory of Random Sets. Springer, London.Google Scholar
Quintanilla, J. A. (2008). Necessary and sufficient conditions for the two-point phase probability function of two-phase random media. Proc. R. Soc. London A 464, 1761–1779.Google Scholar
Rataj, J. (2014). Random sets of finite perimeter. Math. Nachr. 288, 1047–1056.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Shepp, L. A. (1963). On positive-definite functions associated with certain stochastic processes. Tech. Rep. 63-1213-11, Bell Laboratories.Google Scholar
Straka, F. and Štěpán, J. (1988). Random sets in [0,1]. In Transactions of the Tenth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Vol. B, Reidel, Dordrecht, pp. 349–356.Google Scholar
Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York.Google Scholar