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The time of completion of a linear birth-growth model

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

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.

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