The Annals of Applied Probability

Central limit theory for the number of seeds in a growth model in $\bold R\sp d$ with inhomogeneous Poisson arrivals

S. N. Chiu and M. P. Quine

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Abstract

A Poisson point process $\Psi$ 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 $\mathbb{R}^d$ at times $t_i \epsilon [0, \infty)$. Once a seed is born, it begins to create a cell by growing radially in all directions with speed $v > 0$. Points of $\Psi$ 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 $d \geq 1$, we give conditions for asymptotic normality. Rates of convergence are given in all cases.

Article information

Source
Ann. Appl. Probab., Volume 7, Number 3 (1997), 802-814.

Dates
First available in Project Euclid: 16 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1034801254

Digital Object Identifier
doi:10.1214/aoap/1034801254

Mathematical Reviews number (MathSciNet)
MR1459271

Zentralblatt MATH identifier
0888.60016

Subjects
Primary: 60G55: Point processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems
Secondary: 60G60: Random fields 60F17: Functional limit theorems; invariance principles

Keywords
Birth-growth inhomogeneous Poisson process $\mathbb{R}^d$ central limit theorem Brownian motion rate of convergence

Citation

Chiu, S. N.; Quine, M. P. Central limit theory for the number of seeds in a growth model in $\bold R\sp d$ with inhomogeneous Poisson arrivals. Ann. Appl. Probab. 7 (1997), no. 3, 802--814. doi:10.1214/aoap/1034801254. https://projecteuclid.org/euclid.aoap/1034801254


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