Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T10:00:06.543Z Has data issue: false hasContentIssue false
  • English
  • Français

Central limit theorem for a class of random measures associated with germ-grain models

Part of: Limit theorems Stochastic processes General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Lothar Heinrich  and
Ilya S. Molchanov
Lothar Heinrich*
Affiliation:
University of Augsburg
Ilya S. Molchanov*
Affiliation:
University of Glasgow
*
∗ Postal address: Institut für Mathematik, Universität Augsburg, D-86135 Augsburg, Germany.
∗∗ Postal address: Department of Statistics, University of Glasgow, Glasgow G12 8QW, Scotland, U.K. Email address: ilya@stats.gla.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝd generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

Keywords

β-mixingBoolean modelgerm-grain modelintrinsic volumesm-dependent random fieldrandom measurerandom setweak dependence

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 60F05: Central limit and other weak theorems 60G57: Random measures
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 283 - 314
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baddeley, A. J. (1980). A limit theorem for statistics of spatial data. Adv. Appl. Prob. 12, 447–461.CrossRefGoogle Scholar
Benes, V., (1995). On second-order formulas in anisotropic stereology. Adv. Appl. Prob. 27, 326–343.CrossRefGoogle Scholar
Benes, V., Chadœuf, J. and Ohser, J. (1994). On some characteristics of anisotropic fibre processes. Math. Nachr. 169, 5–17.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Doukhan, P. (1995). Mixing: Properties and Examples. (Lecture Notes in Statist. 85.) Springer, Berlin.Google Scholar
Götze, F., Heinrich, L. and Hipp, C. (1995). m-dependent random fields with analytical cumulant generating function. Scand. J. Statist. 22, 183–195.Google Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.Google Scholar
Hanisch, K.-H. (1981). On classes of random sets and point processes. Serdica 7, 160–166.Google Scholar
Heinrich, L. (1988). Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary Poisson cluster process. Math. Nachr. 136, 131–148.CrossRefGoogle Scholar
Heinrich, L. (1988). Asymptotic Gaussianity of some estimators for reduced factorial moment measures and product densities of stationary Poisson cluster processes. Statistics 19, 87–106.CrossRefGoogle Scholar
Heinrich, L. (1992). On existence and mixing properties of germ-grain models. Statistics 23, 271–286.Google Scholar
Heinrich, L. (1993). Asymptotic properties of minimum contrast estimators for parameters of Boolean models. Metrika 31, 349–360.Google Scholar
Heinrich, L. (1994). Normal approximation for some mean-value estimates of absolutely regular tessellations. Math. Meth. Statist. 3, 1–24.Google Scholar
Ivanoff, B. G. (1982). Central limit theorems for point processes. Stochast. Process. Appl. 12, 171–186.Google Scholar
Jolivet, E. (1982). Central limit theorem and convergence of empirical processes for stationary point processes. In Point Processes and Queuing Problems, eds. Bartfai, P. and Tomko, J.. North-Holland, Amsterdam, pp. 117–161.Google Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Academic Press, New York.CrossRefGoogle Scholar
Mase, S. (1982). Asymptotic properties of stereological estimators of volume fraction for stationary random sets. J. Appl. Prob. 19, 111–126.CrossRefGoogle Scholar
Mase, S. (1982). Properties of fourth-order, strong mixing rates and its application to random set theory. J. Multivariate Anal. 12, 549–561.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Molchanov, I. S. (1995). Statistics of the Boolean model: from the estimation of means to the estimation of distributions. Adv. Appl. Prob. 27, 63–86.Google Scholar
Molchanov, I. S. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester.Google Scholar
Molchanov, I. S. and Stoyan, D. (1994). Asymptotic properties of estimators for parameters of the Boolean model. Adv. Appl. Prob. 26, 301–323.Google Scholar
Nguyen, X. X. and Zessin, H. (1979). Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitsth. 48, 133–158.Google Scholar
Schneider, R. (1980). Parallelmengen mit Vielfachheit und Steiner-Formeln. Geom. Dedicata 2, 111–127.Google Scholar
Schneider, R. (1993). Convex Bodies. The Brunn–Minkowski Theory. CUP, Cambridge.CrossRefGoogle Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. Wiley, Chichester.Google Scholar
Vitale, R. A. (1988). An alternate formulation of mean value for random geometric figures. J. Microscopy 151, 197–204.Google Scholar
Volkonskii, V. A. and Rozanov, Yu. A. (1959). Some limit theorems for random functions. Theory Prob. Appl. 4, 178–197.CrossRefGoogle Scholar
Weil, W. (1983). Stereology: a survey for geometers. In Convexity and Its Applications, ed. Gruber, P. M. and Wills, J. M.. Birkhäuser, Basel, pp 360–412.Google Scholar
Weil, W. (1988). Expectation formulas and isoperimetric properties for non-isotropic Boolean models. J. Microscopy 151, 235–245.Google Scholar
Weil, W. (1995). The estimation of mean shape and mean particle number in overlapping particle systems in the plane. Adv. Appl. Prob. 27, 102–119.CrossRefGoogle Scholar
Weil, W. and Wieacker, J. A. (1984). Densities for stationary random sets and point processes. Adv. Appl. Prob. 16, 324–346.CrossRefGoogle Scholar
Yoshihara, K.-I. (1976). Limiting behavior of U-statistics for stationary absolutely regular processes. Z. Wahrscheinlichkeitsth. 35, 237–252.CrossRefGoogle Scholar
Zähle, M., (1982). Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108, 49–72.CrossRefGoogle Scholar
Zähle, M., (1986). Curvature measures and random sets, II. Prob. Theory Rel. Fields 71, 37–58.Google Scholar