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A general linear birth and growth model

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

M. P. Quine  and
W. Szczotka
M. P. Quine*
Affiliation:
University of Sydney
W. Szczotka*
Affiliation:
University of Wrocław
*
∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email address: malcolmq@maths.usyd.edu.au
∗∗ Postal address: Institute of Mathematics, Wrocław University, 50-384 Wrocław, Poland.
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Abstract

We consider a birth and growth model where points (‘seeds’) arrive on a line randomly in time and space and proceed to ‘cover’ the line by growing at a uniform rate in both directions until an opposing branch is met; points which arrive on covered parts of the line do not contribute to the process. Existing results concerning the number of seeds assume that points arrive according to a Poisson process, homogeneous on the line, but possibly inhomogeneous in time. We derive results under less stringent assumptions, namely that the arrival process be a stationary simple point process.

Keywords

Birth and growth on ℝnumber of seedslimit resultsstationary simple point process

MSC classification

Primary: 60G55: Point processes
Secondary: 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 1027 - 1050
Copyright
Copyright © Applied Probability Trust 2000 

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References

Brandt, A., Franken, P. and Lisek, B. (1990). Stationary Stochastic Models. John Wiley, New York.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Chiu, S. N. and Quine, M. P. (1977). 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
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
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
Lai, T. L. (1977). Convergence rates and r-quick versions of the strong law for stationary mixing sequences. Ann. Prob. 5, 693–706.Google Scholar
Meiering, 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
Oodaira, H. and Yoshihara, K.-I. (1971). Note on the law of the iterated logarithm for stationary processes satisfying mixing conditions. Kodai Math. Sem. Rep. 23, 335–342.Google Scholar
Peligrad, M. (1986). Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (a survey). In Dependencies in Probability and Statistics. A Survey of Recent Results. (Oberwolfach 1985), eds Eberlein, E. and Taqqu, M. S.. Birkhäuser, Basel, pp. 193–223.CrossRefGoogle Scholar
Quine, M. P. and Robinson, J. (1990). A linear random growth model. J. Appl. Prob. 27, 499–509.Google Scholar
Szczotka, W. (1986). Joint distribution of waiting time and queue size for single server queues. Dissertationes Math. (Rozprawy Mat.) 248, 1–53.Google Scholar
Vanderbei, R. J. and Shepp, L. A. (1988). A probabilistic model for the time to unravel a strand of DNA. Comm. Statist. Stoch. Models, 4, 299–314.Google Scholar
Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Operat. Res., 5, 67–85.Google Scholar
Yokoyama, R. (1980). Moment bounds for stationary mixing sequences. Z. Wahrscheinlichkeitsth. 53, 45–57.Google Scholar