Central limit theory for the number of seeds in a growth model in ℝ^d with inhomogeneous Poisson arrivals

A Poisson point process ψ 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 ℝ^d at times t_i ∈ [0, ∞). Once a seed is born, it begins to create a cell by growing radially in all directions with speed v > 0. Points of ψ 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.