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Central limit theorem for a birth–growth model with poisson arrivals and random growth speed

Part of: Geometric probability and stochastic geometry Stochastic processes Limit theorems

Published online by Cambridge University Press:  19 January 2024

Chinmoy Bhattacharjee Open the ORCID record for Chinmoy Bhattacharjee [Opens in a new window] ,
Ilya Molchanov  and
Riccardo Turin Open the ORCID record for Riccardo Turin [Opens in a new window]
Chinmoy Bhattacharjee*
Affiliation:
University of Hamburg
Ilya Molchanov*
Affiliation:
University of Bern
Riccardo Turin*
Affiliation:
Swiss Re
*
*Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany. Email address: chinmoy.bhattacharjee@uni-hamburg.de
**Postal address: IMSV, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland. Email address: ilya.molchanov@unibe.ch
***Postal address: Swiss Re Management Ltd, Mythenquai 50/60, 8022 Zurich, Switzerland. Email address: Riccardo_Turin@swissre.com
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Abstract

We consider Gaussian approximation in a variant of the classical Johnson–Mehl birth–growth model with random growth speed. Seeds appear randomly in $\mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The locations, birth times, and growth speeds of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed, the time distribution, and a weight function $h\;:\;\mathbb{R}^d \times [0,\infty) \to [0,\infty)$, we prove a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Such models have previously been considered, albeit with fixed growth speed. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.

Keywords

Spatial birth growthinhomogeneous Poisson processJohnson–Mehl tessellationstabilizationgrowth frontierexposed seeds

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 29
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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