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The time of completion of a linear birth-growth model

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

Published online by Cambridge University Press:  19 February 2016

S. N. Chiu  and
C. C. Yin
S. N. Chiu*
Affiliation:
Hong Kong Baptist University
C. C. Yin*
Affiliation:
Hong Kong Baptist University and Qufu Normal University
*
∗ Postal address: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
∗∗ Email address: [email protected]
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Abstract

Consider the following birth-growth model in ℝ. Seeds are born randomly according to an inhomogeneous space-time Poisson process. A newly formed point immediately initiates a bi-directional coverage by sending out a growing branch. Each frontier of a branch moves at a constant speed until it meets an opposing one. New seeds continue to form on the uncovered parts on the line. We are interested in the time until a bounded interval is completely covered. The exact and limiting distributions as the length of interval tends to infinity are obtained for this completion time by considering a related Markov process. Moreover, some strong limit results are also established.

Keywords

Completion timecoverageinhomogeneous Poisson processJohnson-Mehl modellinear birth-growth modelMarkov processstrong limit theorem

MSC classification

Primary: 60G55: Point processes 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 620 - 627
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

Research supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKBU/2075/98P) and also by the National Natural Science Foundation of China.

References

[1] Chiu, S. N. (1995). Limit theorems for the time of completion of Johnson–Mehl tessellations. Adv. Appl. Prob. 27, 889–910.Google Scholar
[2] Chiu, S. N. (1997). A central limit theorem for linear Kolmogorov's birth–growth models. Stoch. Proc. Appl. 66, 97–106.Google Scholar
[3] Chiu, S. N. and Quine, M. P. (1997). Central limit theory for the number of seeds in a growth model in Rd with inhomogeneous Poisson arrivals. Ann. Appl. Prob. 7, 802–814.Google Scholar
[4] 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
[5] Erhardsson, T. (1996). On the number of high excursions of linear growth processes. Stoch. Proc. Appl. 65, 31–53.Google Scholar
[6] Holst, L., Quine, M. P. and Robinson, J. (1996). A general stochastic model for nucleation and linear growth. Ann. Appl. Prob. 6, 903–921.Google Scholar
[7] 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
[8] Kolmogorov, A. N. (1939). On statistical theory of metal crystallisation. Izv. Akad. Nauk SSSR, Ser. Mat. 3, 355–360.Google Scholar
[9] Meijering, J. L. (1953). Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rept 8, 270–290.Google Scholar
[10] Quine, M. P. and Robinson, J. (1990). A linear random growth model. J. Appl. Prob. 27, 499–509.Google Scholar
[11] Quine, M. P. and Robinson, J. (1992). Estimation for a linear growth model. Statist. Prob. Lett. 15, 295–297.Google Scholar
[12] Vanderbei, R. J. and Shepp, L. A. (1988). A probabilistic model for the time to unravel a strand of DNA. Commun. Statist.–Stoch. Models 4, 299–414.Google Scholar
[13] Wolk, C. P. (1975). Formation of one-dimensional patterns by stochastic processes and by filamentous blue-green algae. Devel. Biol. 46, 370–382.Google Scholar