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.
Publié le : 1997-08-14
Classification:
Birth-growth,
inhomogeneous Poisson process,
$\mathbb{R}^d$,
central limit theorem,
Brownian motion,
rate of convergence,
60G55,
60D05,
60F05,
60G60,
60F17
@article{1034801254,
author = {Chiu, S. N. and Quine, M. P.},
title = {Central limit theory for the number of seeds in a growth model in
$\bold R\sp d$ with inhomogeneous Poisson arrivals},
journal = {Ann. Appl. Probab.},
volume = {7},
number = {1},
year = {1997},
pages = { 802-814},
language = {en},
url = {http://dml.mathdoc.fr/item/1034801254}
}
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., Tome 7 (1997) no. 1, pp. 802-814. http://gdmltest.u-ga.fr/item/1034801254/