The Annals of Applied Probability

Large deviations of the empirical volume fraction for stationary Poisson grain models

Lothar Heinrich

Full-text: Open access


We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function Ln(z)=|Wn|−1logEexp{z|Ξ∩Wn|} of the empirical volume fraction |Ξ∩Wn|/|Wn|, where |⋅| denotes the d-dimensional Lebesgue measure. Here Ξ=⋃i≥1i+Xi) denotes a d-dimensional Poisson grain model (also known as a Boolean model) defined by a stationary Poisson process Πλ=∑i≥1δXi with intensity λ>0 and a sequence of independent copies Ξ12,… of a random compact set Ξ0. For an increasing family of compact convex sets {Wn, n≥1} which expand unboundedly in all directions, we prove the existence and analyticity of the limit lim n→∞Ln(z) on some disk in the complex plane whenever Eexp{a0|}<∞ for some a>0. Moreover, closely connected with this result, we obtain exponential inequalities and the exact asymptotics for the large deviation probabilities of the empirical volume fraction in the sense of Cramér and Chernoff.

Article information

Ann. Appl. Probab., Volume 15, Number 1A (2005), 392-420.

First available in Project Euclid: 28 January 2005

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] 60F10: Large deviations
Secondary: 60G55: Point processes 82B30: Statistical thermodynamics [See also 80-XX]

Poisson grain model with compact grains volume fraction Cox process thermodynamic limit correlation measures cumulants large deviations Berry–Esseen bound Chernoff-type theorem


Heinrich, Lothar. Large deviations of the empirical volume fraction for stationary Poisson grain models. Ann. Appl. Probab. 15 (2005), no. 1A, 392--420. doi:10.1214/105051604000001007.

Export citation


  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Fritz, J. (1970). Generalization of McMillan's theorem to random set functions. Studia Sci. Math. Hungar. \bf5 369–394.
  • Georgii, H.-O. and Zessin, H. (1993). Large deviations and maximum entropy principle for marked point random fields. Probab. Theory Related Fields \bf96 177–204.
  • Götze, F., Heinrich, L. and Hipp, C. (1995). $m$-dependent random fields with analytic cumulant generating function. Scand. J. Statist. \bf22 183–195.
  • Greenberg, W. (1971). Thermodynamic states of classical systems. Comm. Math. Phys. \bf22 259–268.
  • Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.
  • Heinrich, L. (1992). On existence and mixing properties of germ-grain models. Statistics \bf23 271–286.
  • Heinrich, L. and Molchanov, I. S. (1999). Central limit theorem for a class of random measures associated with germ-grain models. Adv. in Appl. Probab. \bf31 283–314.
  • Heinrich, L. and Schmidt, V. (1985). Normal convergence of multidimensional shot noise and rates of this convergence. Adv. in Appl. Probab. \bf17 709–730.
  • Ivanoff, G. (1982). Central limit theorems for point processes. Stochastic Process. Appl. \bf12 171–186.
  • Mase, S. (1982). Asymptotic properties of stereological estimators of volume fraction for stationary random sets. J. Appl. Probab. \bf19 111–126.
  • Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.
  • Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. Wiley, Chichester.
  • Molchanov, I. S. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester.
  • Piau, D. (1999). Large deviations and Young measures for a Poissonian model of biphased material. Ann. Appl. Probab. \bf9 706–718.
  • Rota, G.-C. (1964). On the foundations of combinatorial theory: I. Theory of Möbius functions. Z. Wahrsch. Verw. Gebiete \bf2 340–368.
  • Ruelle, D. (1964). Cluster properties of correlation functions of classical gases. Rev. Mod. Phys. \bf36 580–586.
  • Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. Benjamin, Amsterdam.
  • Saulis, L. and Statulevičius, V. (1991). Limit Theorems for Large Deviations. Kluwer Academic, Dordrecht.
  • Statulevičius, V. A. (1966). On large deviations. Z. Wahrsch. Verw. Gebiete \bf6 133–144.
  • Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Wiley, Chichester.