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

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.