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A linear random growth model

Published online by Cambridge University Press:  14 July 2016

M. P. Quine  and
J. Robinson
M. P. Quine
Affiliation:
University of Sydney
J. Robinson*
Affiliation:
University of Sydney
*
∗ Postal address for both authors: Department of Mathematical Statistics, University of Sydney, NSW 2006, Australia.
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Abstract

Points start to form on an ‘uncovered' unit interval according to a Poisson process with parameter λ. From newly formed points a covering region grows in both directions at velocity v, while new points continue to form on uncovered parts of the interval. Eventually the whole interval will be covered. Let N ≧ 1 denote the total number of points formed. We derive integral expressions for E(N) and Var(N) and give precise asymptotic expressions for these moments as ρ = λ/v →∞. Asymptotic normality of N is also established.

Keywords

POISSON PROCESS WITH INHIBITIONGROWTHMOMENTSCENTRAL LIMIT THEOREM

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 3 , September 1990 , pp. 499 - 509
Copyright
Copyright © Applied Probability Trust 1990 

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References

Bennett, M. R. and Robinson, J. (1990) Probabilistic secretion of quanta at the neuromuscular junction: non-uniformity, autoreceptors and the binomial hypothesis. Proc. R. Soc. London B. To appear.Google Scholar
Chung, K. L. (1974) A Course in Probability Theory, 2nd edn. Academic Press, New York.Google Scholar
Gilbert, E. N. (1967) Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.CrossRefGoogle Scholar
Meiering, 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
Shimizu, R. and Davies, L. (1981) General characterization theorems for the Weibull and the stable distributions. Sankhya A43, 282–310.Google Scholar
Vanderbei, R. J. and Shepp, L. A. (1988) A probabilistic model for the time to unravel a strand of DNA. Commun. Statist. – Stochastic Models 4, 299–314.Google Scholar
Wolk, C. P. (1975) Formation of one-dimensional patterns by stochastic processes and by filamentous blue-green algae. Dev. Biol. 46, 370–382.Google Scholar