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Compound Poisson approximation for the Johnson-Mehl model

Part of: Geometric probability and stochastic geometry Markov processes 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: ter@math.kth.se
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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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