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The variance of a Poisson process of domains

Published online by Cambridge University Press:  14 July 2016

A. M. Kellerer
A. M. Kellerer*
Affiliation:
University of Würzburg
*
∗Postal address: Institut für Med. Strahlenkunde der Universität Würzburg, Versbacher Str. 5, D-8700 Würzburg, W. Germany.
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Abstract

A familiar relation links the densities that result for the intersection of a convex body and straight lines under uniform isotropic randomness with those that result under weighted randomness. An extension of this relation to the intersection of more general domains is utilized to obtain the variance of the n-dimensional measure of the intersection of two bodies under uniform isotropic randomness. The formula for the variance contains the point-pair-distance distributions for the two domains — or the closely related geometric reduction factors. The result is applied to derive the variance of the intersection of a Boolean scheme, i.e. a stationary, isotropic Poisson process of domains, with a fixed sampling region.

Keywords

RANDOM INTERSECTION OF GEOMETRIC OBJECTSVARIANCE OF OVERLAPLATTICE PROBLEMPOISSON PROCESSBOOLEAN SCHEMEPOINT-PAIR-DISTANCE DISTRIBUTION
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 2 , June 1986 , pp. 307 - 321
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Work supported by Euratom Contract BI-6-0013 D(B) and Contract 96731 with GSI (Gesellschaft für Schwerionenforschung, Darmstadt).

References

[1] Berger, M. J. (1970) Beta-ray dosimetry calculations with the use of point kernels. In Medical Radionuclides: Radiation Dose and Effects, ed. Cloutier, R. J. et al., 63–86. (Also available as USAEC Report CONF-691212 from the National Technical Information Service, Springfield, VA 22161, USA).Google Scholar
[2] Blaschke, W. (1937) Integralgeometrie 21. Über Schiebungen. Math. Z. 42, 399–410.CrossRefGoogle Scholar
[3] Bronowski, J. and Neyman, J. (1944) The variance of the measure of a two-dimensional random set. Ann. Math. Statist. 16, 330–341.Google Scholar
[4] Coleman, R. (1979) An Introduction to Mathematical Stereology. Memoir Series, Dept of Theoretical Statistics, Institute of Mathematics, University of Aarhus.Google Scholar
[5] Davy, P. J. (1976) Projected thick sections through multidimensional particle aggregates. J. Appl. Prob. 13, 714–722.Google Scholar
[6] Enns, E. G. and Ehlers, P. F. (1978) Random paths through a convex region. J. Appl. Prob. 15, 144–152.CrossRefGoogle Scholar
[7] Garwood, F. (1947) The variance of the overlap of geometrical figures with reference to a bombing problem. Biometrika 14, 1–17.Google Scholar
[8] Kellerer, A. M. (1981) Proximity functions for general right cylinders. Radiation Res. 86, 264–276.CrossRefGoogle Scholar
[9] Kellerer, A. M. (1983) On the number of clumps resulting from the overlap of randomly placed figures in a plane. J. Appl. Prob. 20, 126–135.Google Scholar
[10] Kellerer, A. M. (1984) Chord-length distributions and related quantities for spheroids. Radiation Res. 98, 425–437.Google Scholar
[11] Kellerer, H. G. (1984) Minkowski functionals of Poisson processes. Z. Wahrscheinlichkeitsth. 67, 63–84.CrossRefGoogle Scholar
[12] Kendall, D. G. (1948) On the number of lattice points inside a random oval. Quart. J. Math. (Oxford) 19, 1–26.CrossRefGoogle Scholar
[13] Kendall, D. G. and Rankin, R. A. (1953) On the number of points of a given lattice in a random hypersphere. Quart. J. Math. (2) 4, 178–189.Google Scholar
[14] Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
[15] Kingman, J. F. C. (1965) Mean free paths in a convex reflecting region. J. Appl. Prob. 2, 162–168.CrossRefGoogle Scholar
[16] Matheron, G. (1967) Eléments pour une theorie des milieux poreux. Masson, Paris.Google Scholar
[17] Matheron, G. (1974) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
[18] Miles, R. E. (1979) Some new integral geometric formulae, with stochastic applications. J. Appl. Prob. 16, 592–606.CrossRefGoogle Scholar
[19] Piefke, F. (1978) Beziehungen zwischen der Sehnenlängenverteilung und der Verteilung des Abstandes zweier zufälliger Punkte im Eikörper. Z. Wahrscheinlichkeitsth. 43, 129–134.CrossRefGoogle Scholar
[20] Pitts, E. (1981) The overlap of random particles and similar problems: expressions for variance of coverage and its analogue. SIAM J. Appl. Math. 41, 493–498.Google Scholar
[21] Robbins, H. E. (1944) On the measure of a random set. Ann. Math. Statist. 15, 70–74.Google Scholar
[22] Robbins, H. E. (1945) On the measure of a random set, II. Ann. Math. Statist. 16, 342–347.CrossRefGoogle Scholar
[23] Russell, A. M. and Josephson, N. S. (1965) Measurement of area by counting. J. Appl. Prob. 2, 339–351.CrossRefGoogle Scholar
[24] Santaló, L. (1947) On the first two moments of the measure of a random set. Ann. Math. Statist. 18, 37–49.Google Scholar
[25] Santaló, L. (1953) Introduction to Integral Geometry. Herman, Paris.Google Scholar
[26] Weil, W. (1984) Densities of Quermassintegrals for stationary random sets. In Stochastic Geometry, Geometric Statistics, Stereology ed. Ambartzumian, R. and Weil, W., Teubner Texte zur Mathematik 35, Teubner, Leipzig, 233–247.Google Scholar