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Refined distributional approximations for the uncovered set in the Johnson-Mehl model

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  • Erhardsson, Torkel

Abstract

Let [Phi]z be the uncovered set (i.e., the complement of the union of intervals) at time z in the one-dimensional Johnson-Mehl model. We derive a bound for the total variation distance between the distribution of the number of components of [Phi]z[intersection](0,t] and a compound Poisson-geometric distribution, which is sharper and simpler than an earlier bound obtained by Erhardsson. We also derive a previously unavailable bound for the total variation distance between the distribution of the Lebesgue measure of [Phi]z[intersection](0,t] and a compound Poisson-exponential distribution. Both bounds are O(z[beta](t)/t) as t-->[infinity], where z[beta](t) is defined so that the expected number of components of [Phi]z[beta](t)[intersection](0,t] converges to [beta]>0 as t-->[infinity], and the parameters of the approximating distributions are explicitly calculated.

Suggested Citation

  • Erhardsson, Torkel, 2001. "Refined distributional approximations for the uncovered set in the Johnson-Mehl model," Stochastic Processes and their Applications, Elsevier, vol. 96(2), pages 243-259, December.
    • Handle: RePEc:eee:spapps:v:96:y:2001:i:2:p:243-259
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    References listed on IDEAS

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    1. Chiu, S. N., 1997. "A central limit theorem for linear Kolmogorov's birth-growth models," Stochastic Processes and their Applications, Elsevier, vol. 66(1), pages 97-106, February.
    2. Erhardsson, Torkel, 1996. "On the number of high excursions of linear growth processes," Stochastic Processes and their Applications, Elsevier, vol. 65(1), pages 31-53, December.
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