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Limit theorems for the time of completion of Johnson-Mehl tessellations

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

Published online by Cambridge University Press:  01 July 2016

S. N. Chiu
S. N. Chiu*
Affiliation:
Freiberg University of Mining and Technology
*
* Present address: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
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Abstract

Johnson–Mehl tessellations can be considered as the results of spatial birth–growth processes. It is interesting to know when such a birth–growth process is completed within a bounded region. This paper deals with the limiting distributions of the time of completion for various models of Johnson–Mehl tessellations in ℝd and k-dimensional sectional tessellations, where 1 ≦ k < d, by considering asymptotic coverage probabilities of the corresponding Boolean models. Random fractals as the results of birth–growth processes are also discussed in order to show that a birth–growth process does not necessarily lead to a Johnson–Mehl tessellation.

Keywords

BOOLEAN MODELSCOVERAGEEXTREME VALUE DISTRIBUTIONSJOHNSON–MEHL TESSELLATIONSFRACTALSSTOCHASTIC GEOMETRY

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 4 , December 1995 , pp. 889 - 910
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Supported by a scholarship from DAAD, Postfach 200 404, D-53134 Bonn, Germany.

References

Chiu, S. N. (1994) Limit theorem for the time of completion of Johnson-Mehl tessellations. Technical Report, Institut für Stochastik, TU Bergakademie Freiberg.Google Scholar
Cowan, R., Chiu, S. N. and Holst, L. (1995) A limit theorem for the replication time of a DNA molecule. J. Appl. Prob. 32, 296–303.CrossRefGoogle Scholar
Evans, U. R. (1945) The laws of expanding circles and spheres in relation to the lateral growth of surface films and the grain size of metals. Trans. Faraday Soc. 41, 365–374.Google Scholar
Frost, H. J. and Thompson, C. V. (1987) The effect of nucleation conditions on the topology and geometry of two-dimensional grain structures. Acta Metallurgica 35, 529–540.Google Scholar
Gilbert, E. N. (1962) Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.CrossRefGoogle Scholar
Hall, P. (1985a) Distribution of size, structure and number of vacant regions in a high-intensity mosaic. Z. Wahrscheinlichkeitsth. 70, 237–261.Google Scholar
Hall, P. (1985b) On the coverage of k-dimensional space by k-dimensional spheres. Ann. Prob. 13, 991–1002.Google Scholar
Hall, P. (1988) Introduction to the Theory of Coverage Processes. Wiley, New York.Google Scholar
Johnson, W. A. and Mehl, R. F. (1939) Reaction kinetics in processes of nucleation and growth. Trans. Amer. Inst. Min. Metal. Petro. Eng. 135, 410–458.Google Scholar
Kolmogorov, A. N. (1937) On statistical theory of metal crystallization. Izvestia Academy of Science, USSR, ser. Math. 3, 355–360 (in Russian).Google Scholar
Krickeberg, K. (1972) Theory of hyperplane processes. In Stochastic Point Processes, ed. Lewis, P. A. W., pp. 514–521. Wiley-Interscience, New York.Google Scholar
Leadbetter, M. R. (1983) Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitsth. 65, 291–306.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Problems of Random Sequences and Processes. Springer-Verlag, Berlin.Google Scholar
Leadbetter, M. R. and Rootzen, H. (1988) Extremal theory for stochastic processes. Ann. Prob. 16, 431–478.CrossRefGoogle Scholar
Mandelbrot, B. B. (1972) Renewal sets and random cutouts. Z. Wahrscheinlichkeitsth. 22, 145–157.CrossRefGoogle Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Meijering, J. L. (1953) Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270–290.Google Scholar
Miles, R. E. (1972) The random division of space. Adv. Appl. Prob. Suppl. 4, 243–266.CrossRefGoogle Scholar
Møller, J. (1992) Random Johnson-Mehl tessellations. Adv. Appl. Prob. 24, 814–844.Google Scholar
Møller, J. (1994a) Generation of Johnson-Mehl crystals and comparative analysis of models for random nucleation. Research Report 275, Department of Theoretical Statistics, University of Aarhus.Google Scholar
Møller, J. (1994b) Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics 87, Springer-Verlag, New York.Google Scholar
Okabe, A., Boots, B. N. and Sugihara, K. (1992) Spatial Tessellations, Concepts and Applications of Voronoi Diagrams. Wiley, New York.Google Scholar
Quine, M. P. and Robinson, J. (1990) A linear random growth model. J. Appl. Prob. 27, 499–509.Google Scholar
Quine, M. P. and Robinson, J. (1992) Estimation for a linear growth model. Statist. Prob. Lett. 15, 293–297.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
Shepp, L. A. (1972) Covering the line with random intervals. Z. Wahrscheinlichkeitsth. 23, 163–170.Google Scholar
Solomon, H. (1978) Geometric Probability. SIAM, Philadelphia.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and Its Applications. Akademie-Verlag, Berlin.Google Scholar
Vanderbei, R. J. and Shepp, L. A. (1988) A probabilistic model for the time to unravel a strand of DNA. Stoch. Models 4, 299–314.Google Scholar
Zähle, U. (1984) Random fractals generated by random cutouts. Math. Nachr. 116, 27–52.Google Scholar