A Poisson point process 9 in d-dimensional Euclidean space and in time is used to generate a birth–growth model: seeds are born randomly at locations xi in Rd at times [formula]. Once a seed is born, it begins to create a cell by growing radially in all directions with speed v > 0. Points of 9 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.
- inhomogeneous Poisson process
- R d
- central limit theorem
- Brownian motion
- rate of convergence