No abstract

A Poisson point process 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 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.

SummaryWe consider the limiting behaviour of a k-type (k < ∞) Galton-Watson process which is augmented at each generation by a stochastic immigration component. In Section 2, conditions for ergodicity are found for a subclass of such processes. In Section 3, expressions are derived for the first two moments of the nth generation (by way of a recurrence relation) and for the first two asymptotic moments, in a manner which to some extent generalises previous results.

A known gamma-type result for the Poisson process states that certain domains defined through configuration of the points (or ‘particles’) of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples—based on the Voronoi and Delaunay tessellations—where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.

The model considered here has arisen in a number of completely separate contexts: release of neurotransmitter at neuromuscular synapses, unravelling of strands of DNA, differentiation of cells into heterocysts in algae and growth of crystals. After a shear transformation the model becomes a Markov process, based on a Poisson process on the upper half plane, homogeneous in the horizontal (time) direction, which increases at unit rate except for occasional "drops." By considering the process separately when it is above or below a given "level," for any interval on the time axis, we obtain in particular exact moment results and prove asymptotic normality for long time intervals for the number of downcrossings in the interval, the total time in the interval when the process is below the specified level and the number of drops in the interval. Limit distributions as the length of interval tends to infinity are obtained for the level at which the interval is "covered." It is shown that several problems considered in the literature have analytic solutions as special cases of the general model. The numerical results from one special case are compared to statistics obtained from experimental data from neurobiology.