## Abstract

A Poisson point process ψ in d-dimensional Euclidean space and in time is used to generate a birth-growth model: seeds are born randomly at locations x_{i} in ℝ^{d} at times t_{i} ∈ [0, ∞). Once a seed is born, it begins to create a cell by growing radially in all directions with speed v > 0. Points of ψ contained in such cells are discarded, that is, thinned. We study the asymptotic distribution of the number of seeds in a region, as the volume of the region tends to infinity. When d = 1, we establish conditions under which the evolution over time of the number of seeds in a region is approximated by a Wiener process. When d ≥ 1, we give conditions for asymptotic normality. Rates of convergence are given in all cases.

Original language | English |
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Pages (from-to) | 802-814 |

Number of pages | 13 |

Journal | Annals of Applied Probability |

Volume | 7 |

Issue number | 3 |

Publication status | Published - Aug 1997 |

## Scopus Subject Areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

## User-Defined Keywords

- ℝ
- Birth-growth
- Brownian motion
- Central limit theorem
- Inhomogeneous Poisson process
- Rate of convergence

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