Department of Mathematics
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.
Birth–growth, inhomogeneous Poisson process, R d, central limit theorem, Brownian motion, rate of convergence
Source Publication Title
Annals of Applied Probability
Institute of Mathematical Statistics
Research partially supported by an Australian Research Council Institutional Grant.
Link to Publisher's Edition
Chiu, S., & Quine, M. (1997). Central limit theory for the number of seeds in a growth model in Rd with inhomogeneous poisson arrivals. Annals of Applied Probability, 7 (3), 802-814. https://doi.org/10.1214/aoap/1034801254