The Annals of Applied Probability

Regularity conditions in the realisability problem with applications to point processes and random closed sets

Raphael Lachieze-Rey and Ilya Molchanov

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study existence of random elements with partially specified distributions. The technique relies on the existence of a positive extension for linear functionals accompanied by additional conditions that ensure the regularity of the extension needed for interpreting it as a probability measure. It is shown in which case the extension can be chosen to possess some invariance properties.

The results are applied to the existence of point processes with given correlation measure and random closed sets with given two-point covering function or contact distribution function. It is shown that the regularity condition can be efficiently checked in many cases in order to ensure that the obtained point processes are indeed locally finite and random sets have closed realisations.

Article information

Ann. Appl. Probab. Volume 25, Number 1 (2015), 116-149.

First available in Project Euclid: 16 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 28C05: Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures 46A40: Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42] 47B65: Positive operators and order-bounded operators 60G55: Point processes 74A40: Random materials and composite materials 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Point process correlation measure random closed set two-point covering probability contact distribution function realisability


Lachieze-Rey, Raphael; Molchanov, Ilya. Regularity conditions in the realisability problem with applications to point processes and random closed sets. Ann. Appl. Probab. 25 (2015), no. 1, 116--149. doi:10.1214/13-AAP990.

Export citation


  • [1] Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis, 3rd ed. Springer, Berlin.
  • [2] Bourbaki, N. (1989). General Topology. Springer, Berlin. Chapters 5–10. Translated from the French, reprint of the 1966 edition.
  • [3] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • [4] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Univ. Press, Cambridge.
  • [5] Fristedt, B. and Gray, L. (1997). A Modern Approach to Probability Theory. Birkhäuser, Boston, MA.
  • [6] Galerne, B. and Lachièze-Rey, R. (2013). Random measurable sets and realisability problems. Unpublished manuscript.
  • [7] Holley, R. A. and Stroock, D. W. (1978). Nearest neighbor birth and death processes on the real line. Acta Math. 140 103–154.
  • [8] Jiao, Y., Stillinger, F. H. and Torquato, S. (2007). Modeling heterogeneous materials via two-point correlation functions: Basic principles. Phys. Rev. E (3) 76 031110.
  • [9] Kellerer, H. G. (1964). Verteilungsfunktionen mit gegebenen Marginalverteilungen. Z. Wahrsch. Verw. Gebiete 3 247–270.
  • [10] Kondratiev, Yu. G. and Kutoviy, O. V. (2006). On the metrical properties of the configuration space. Math. Nachr. 279 774–783.
  • [11] König, H. (1997). Measure and Integration: An Advanced Course in Basic Procedures and Applications. Springer, Berlin.
  • [12] Kuna, T., Lebowitz, J. L. and Speer, E. R. (2007). Realizability of point processes. J. Stat. Phys. 129 417–439.
  • [13] Kuna, T., Lebowitz, J. L. and Speer, E. R. (2011). Necessary and sufficient conditions for realizability of point processes. Ann. Appl. Probab. 21 1253–1281.
  • [14] Kuratowski, K. (1966). Topology, Vol. I. Academic Press, New York.
  • [15] Lenard, A. (1975). States of classical statistical mechanical systems of infinitely many particles. I. Arch. Ration. Mech. Anal. 59 219–239.
  • [16] Lenard, A. (1975). States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures. Arch. Ration. Mech. Anal. 59 241–256.
  • [17] Markov, K. Z. (1995). On the “triangular” inequality in the theory of two-phase random media. Technical Report 89/1995, 159-166, Annuaire de’Universite de Sofia “St. Klimen Ohridski,” Faculte de mathematiques et Informatique, Livre I—Mathematiques et Mécanique.
  • [18] Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.
  • [19] Matheron, G. (1993). Une conjecture sur covariance d’un ensemble aleatoire. Cahiers de Géostatistique 107 107–113.
  • [20] Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Cambridge Univ. Press, Cambridge.
  • [21] McMillan, B. (1955). History of a problem. J. Soc. Indust. Appl. Math. 3 119–128.
  • [22] Molchanov, I. (2005). Theory of Random Sets. Springer, London.
  • [23] Molchanov, I. S. (1989). On convergence of empirical accompanying functionals of stationary random sets. Theory Probab. Math. Statist. 38 107–109.
  • [24] Quintanilla, J. A. (2008). Necessary and sufficient conditions for the two-point phase probability function of two-phase random media. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464 1761–1779.
  • [25] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • [26] Sharakhmetov, Sh. and Ibragimov, R. (2002). A characterization of joint distribution of two-valued random variables and its applications. J. Multivariate Anal. 83 389–408.
  • [27] Shepp, L. A. (1963). On positive-definite functions associated with certain stochastic processes. Technical Report 63-1213-11, Bell Telephone Laboratories, Murray Hill, NJ.
  • [28] Shepp, L. A. (1967). Covariances of unit processes. In Proc. Working Conf. Stochastic Processes 205–218. Santa Barbara, California, CA.
  • [29] Silverman, R. J. (1956). Invariant linear functions. Trans. Amer. Math. Soc. 81 411–424.
  • [30] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Wiley, Chichester.
  • [31] Torquato, S. (1999). Exact conditions on physically realizable correlation functions of random media. J. Chem. Phys. 111 8832–8837.
  • [32] Torquato, S. (2002). Random Heterogeneous Materials. Springer, New York.
  • [33] Torquato, S. (2006). Necessary conditions on realizable two-point correlation functions of random media. Indus. Eng. Chem. Res. 45 6923–6928.
  • [34] Torquato, S. and Stell, G. (1982). Microstructure of two-phase random media. I. The $n$-point probability functions. J. Chem. Phys. 77 2071–2077.
  • [35] Vulikh, B. Z. (1967). Introduction to the Theory of Partially Ordered Spaces. Wolters-Noordhoff, Groningen.
  • [36] Whittle, P. (1992). Probability via Expectation. Springer, New York.