Central limit theorem for germination-growth models in ℝd with non-Poisson locations

Seeds are randomly scattered in ℝ d according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.

Limit theorems for the time of completion of Johnson-Mehl tessellations

Johnson–Mehl tessellations can be considered as the results of spatial birth–growth processes. It is interesting to know when such a birth–growth process is completed within a bounded region. This paper deals with the limiting distributions of the time of completion for various models of Johnson–Mehl tessellations in ℝdandk-dimensional sectional tessellations, where 1 ≦k<d, by considering asymptotic coverage probabilities of the corresponding Boolean models. Random fractals as the results of birth–growth processes are also discussed in order to show that a birth–growth process does not necessarily lead to a Johnson–Mehl tessellation.

An interference problem with application to crystal growth

We model diffusion-controlled crystal growth as an interference problem. The crystal layers grow by nucleation (initiation of crystallization centers) followed by attachment of molecules to the nucleus. A forming crystal layer completes by either spreading across the length of the crystal or by colliding with another spreading crystal layer. This model differs from the classical Johnson-Mehl-Kolmogorov model in that nucleation happens only on boundaries of a ‘seed’ crystal as opposed to nucleation from random points in a given region. Our results also differ from the limiting results found for this classical model. We use the invariant measure of an embedded Markov process to find the growth rate of the crystal in terms of the nucleation rates. Ergodic theorems are then used to derive explicit formulae for some stationary probabilities.

The time of completion of a linear birth-growth model

Consider the following birth-growth model in ℝ. Seeds are born randomly according to an inhomogeneous space-time Poisson process. A newly formed point immediately initiates a bi-directional coverage by sending out a growing branch. Each frontier of a branch moves at a constant speed until it meets an opposing one. New seeds continue to form on the uncovered parts on the line. We are interested in the time until a bounded interval is completely covered. The exact and limiting distributions as the length of interval tends to infinity are obtained for this completion time by considering a related Markov process. Moreover, some strong limit results are also established.

Nonparametric and parametric estimation for a linear germination-growth model

Seeds are planted on the interval [0, L] at various locations. Each seed has a location x and a potential germination time t epsilon [0, infinity), and it is assumed that the collection of such (x, t) pairs forms a Poisson process in [0, L] x [0, infinity) with intensity measure dxd lambda(t). From each seed that germinates, an inhibiting region grows bidirectionally at rate 2v. These regions inhibit germination of any seed in the region with a later potential germination time. Thus, seeds only germinate in the uninhibited part of [0, L]. We want to estimate lambda on the basis of one or more realizations of the process, the data being the locations and germination times of the germinated seeds. We derive the maximum likelihood estimator of v and a nonparametric estimator of lambda and describe methods of obtaining parametric estimates from it, illustrating these with reference to gamma densities. Simulation results are described and the methods applied to some neurobiological data. An Appendix outlines the S-PLUS code used.

Compound Poisson approximation for the Johnson-Mehl model

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

Compound Poisson approximation for the Johnson-Mehl model

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

Modelling random linear nucleation and growth by a Markov chain

In an attempt to investigate the adequacy of the normal approximation for the number of nuclei in certain growth/coverage models, we consider a Markov chain which has properties in common with related continuous-time Markov processes (as well as being of interest in its own right). We establish that the rate of convergence to normality for the number of ‘drops’ during times 1,2,…n is of the optimal ‘Berry–Esséen’ form, as n → ∞. We also establish a law of the iterated logarithm and a functional central limit theorem.

Modelling random linear nucleation and growth by a Markov chain

In an attempt to investigate the adequacy of the normal approximation for the number of nuclei in certain growth/coverage models, we consider a Markov chain which has properties in common with related continuous-time Markov processes (as well as being of interest in its own right). We establish that the rate of convergence to normality for the number of ‘drops’ during times 1,2,…n is of the optimal ‘Berry–Esséen’ form, as n → ∞. We also establish a law of the iterated logarithm and a functional central limit theorem.