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Compound Poisson approximation for the Johnson-Mehl model

Part of: Markov processes Geometric probability and stochastic geometry Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Torkel Erhardsson
Torkel Erhardsson*
Affiliation:
KTH, Stockholm
*
∗Postal address: Department of Mathematics, KTH, S-10044 Stockholm, Sweden. Email address: [email protected]
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Abstract

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

Keywords

Johnson-Mehl modeluncovered setcompound Poisson approximationerror boundstationary Markov chain‘rare’ set; coupling

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60D05: Geometric probability and stochastic geometry
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 101 - 117
Copyright
Copyright © 2000 by The Applied Probability Trust 

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References

Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
Chiu, S. N. (1995). Limit theorems for the time of completion of Johnson–Mehl tessellations. Adv. Appl. Prob. 27, 889–910.CrossRefGoogle Scholar
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.CrossRefGoogle 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.Google Scholar
Erhardsson, T. (1996). On the number of high excursions of linear growth processes. Stoch. Proc. Appl. 65, 31–53.Google Scholar
Erhardsson, T. (1997). Compound Poisson approximation for Markov chains. Ph.D. Thesis, Royal Institute of Technology, Stockholm.Google Scholar
Erhardsson, T. (1999). Compound Poisson approximation for Markov chains using Stein's method. Ann. Prob. 27, 565–596.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
Kolmogorov, A. N. (1937). On the statistical theory of metal crystallization [In Russian]. Izv. Akad. Nauk SSSR Ser. Mat. 3, 355–360. Translation in: (1992). Selected Works of A. N. Kolmogorov, Vol. II: Probability Theory and Mathematical Statistics, ed. A. N. Shiryayev. Kluwer, Dordrecht, pp. 188–192.Google Scholar
Møller, J. (1992). Random Johnson–Mehl tessellations. Adv. Appl. Prob. 24, 814–844.Google Scholar
Quine, M. P., and Robinson, J. (1990). A linear random growth model. J. Appl. Prob. 27, 499–509.Google Scholar
Stoyan, D., Kendall, W. S., and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, Chichester.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