Lothar Heinrich

NORMAL CONVERGENCE OF MULTIDIMENSIONAL SHOT NOISE AND RATES OF THIS CONVERGENCE

Using a representation formula expressing the mixed cumulants of realvalued random variables by corresponding moments, sufficient conditions are given for the normal convergence of suitably standardized shot noise assuming that the generating stationary point process is independently marked and Brillinger mixing and that its intensity tends to oo. Furthermore, estimates for the rate of this normal convergence are obtained by exploiting a general lemma on probabilities of large deviations and on the rate of normal convergence.

On existence and mixing properties of germ-grain models

We give a rigorous definition of germ-grain models (ggms) which were introduced in [6] as at most countable unions of random closed sets (called grains) in translated by the atoms (called germs) of a point process in , and establish conditions under which the random set Z in a.s. closed. In case of i.i.d. grains we prove a continuity theorem for ggms in terms of weak convergence. Further, we characterize ergodicity and (weak) mixing of stationary ggms with a.s. compact grains by the corresponding properties of the underlying stationary point process. As a consequence we apply an ergodic theorem of Nguyen and Zessin [9] to spatial averages of certain geometric functionals of ggms with a.s. compact convex grains.

Asymptotic properties of minimum contrast estimators for parameters of boolean models

SummaryThis paper presents a method for the estimation of parameters of random closed sets (racs’s) in ℝd based on a single realization within a (large) convex sampling window. The essential idea first applied by Diggle (1981) in a special case consists in defining the estimation by minimizing a suitably defined distance (called contrast function) between the true and the empirical contact distribution function of the racs under consideration, where the most relevant case of Boolean models is discussed in details. The resulting estimates are shown to be strongly consistent (if the racs is ergodic) and asymptotically normal (if the racs is Boolean) when the sampling window expands unboundedly.

Asymptotic Methods in Statistics of Random Point Processes

First we put together basic definitions and fundamental facts and results from the theory of (un)marked point processes defined on Euclidean spaces \({\mathbb{R}}^{d}\). We introduce the notion random marked point process together with the concept of Palm distributions in a rigorous way followed by the definitions of factorial moment and cumulant measures and characteristics related with them. In the second part we define a variety of estimators of second-order characteristics and other so-called summary statistics of stationary point processes based on observations on a “convex averaging sequence” of windows \(\{W_{n},\,n \in \mathbb{N}\}\). Although all these (mostly edge-corrected) estimators make sense for fixed bounded windows our main issue is to study their behaviour when W n grows unboundedly as n → ∞. The first problem of large-domain statistics is to find conditions ensuring strong or at least mean-square consistency as n → ∞ under ergodicity or other mild mixing conditions put on the underlying point process. The third part contains weak convergence results obtained by exhausting strong mixing conditions or even m-dependence of spatial random fields generated by Poisson-based point processes. To illustrate the usefulness of asymptotic methods we give two Kolmogorov–Smirnov-type tests based on K-functions to check complete spatial randomness of a given point pattern in \({\mathbb{R}}^{d}\).

Strong convergence of kernel estimators for product densities of absolutely regular point processes

In this paper we prove almost sure convergence of kernel-type estimators of second-order product densities for stationary absolutely regular (β-mixing) point processes in R d . This type of mixing condition can be verified for various classes of point processes under mild additional assumptions. We also obtain rates of convergence which mainly depend on the decay of the mixing coefficient and the choice of the bandwidth. In case of motion-invariant processes the behaviour of kernel estimators of the pair correlation function is considered separately. The results are applied to kernel-type renewal density estimators.

Weak and strong convergence of empirical distribution functions from germ-grain processes

We observe randomly placed random compact sets (called grains or particles) in a bounded, convex sampling window W n of the d-dimensional Euclidean space which is assumed to expand unboundedly in all directions as n→∞. In addition, we suppose that the grains are independent copies of a so-called typical grain Ξ0, which are shifted by the atoms of a homogeneous point process Ψ in such a way that each individual grain lying within W n can be observed. We define an appropriate estimator ˆ F n (t) for the distribution function F(t) of some m-dimensional vector f(Ξ0)=(f 1(Ξ0), ˙s, f m (Ξ0)) (describing shape and size of Ξ0) on the basis of the corresponding data vectors of those grains which are completely observable in W n . As main results, we prove a Glivenko-type theorem for ˆ F n (t) and the weak convergence of the multivariate empirical processes √Ψ(W n )(ˆ F n (t)−F(t)) to an m-dimensional Brownian bridge process as n→∞.

Numerical and Analytical Computation of Some Second-Order Characteristics of Spatial Poisson-Voronoi Tessellations

We describe and discuss the explicit calculation of the pair correlation function of the point process of nodes associated with a three-dimensional stationary Poisson – Voronoi tessellation. Moreover, the precise asymptotics for the variance of the number of nodes in an expanding region and the variance of the number of vertices of the typical Poisson – Voronoi polyhedron are obtained. This gives rise to an asymptotically exact confidence interval for the number of nodes and cells when the sampling region is large enough. A geometric interpretation of our formulae shows that, among others, an essential problem is to calculate the mean volume of a tetrahedron whose vertices are uniformly distributed on a circular domain of the unit sphere.

LIMIT THEOREMS FOR STATIONARY TESSELLATIONS WITH RANDOM INNER CELL STRUCTURES

We consider stationary and ergodic tessellations X = Ξ n n≥1 in R d , where X is observed in a bounded and convex sampling window W p ⊂ R d . It is assumed that the cells Ξ n of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J 0 acting on R d . In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in W p , as W p ↑ R d . It turns out that this functional provides an unbiased estimator for the intensity vector associated with J 0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.

Kernel estimation of the diameter distribution in boolean models with spherical grains

We investigate asymptotic properties Including MSE of kernel estimators of the second-order product density of the point process of ‘exposed tangent points’ (for given direction ) associated with a stationary d-dimensional Boolean model with convex compact grains. Under minimal conditions on the typical grain we prove that the square root of the kernel estimators is asymptotically normally distributed with constant variance which only depends on the chosen kernel function. In the particular case of spherical grains, as first shown by Molchanov and Stoyan (1994), the diameter distribution function F(t) is just equal to the product of and some function of t>=0 which can be estimated by standard methods. Using this fact we are able to derive a multivariate CLT for a suitably defined empirical diameter distribution function. Owing to this result we suggest a χ2 ‐ goodness-of-fit-test for testing a hypothetical diameter distribution.

Asymptotic goodness-of-fit tests for the Palm mark distribution of stationary point processes with correlated marks

We consider spatially homogeneous marked point patterns in an unboundedly expanding convex sampling window. Our main objective is to identify the distribution of the typical mark by constructing an asymptotic � 2 -goodness-of-fit test. The corresponding test statistic is based on a natural empirical version of the Palm mark distribution and a smoothed covariance estimator which turns out to be mean-square consistent. Our approach does not require independent marks and allows dependences between the mark field and the point pattern. Instead we impose a suitable �-mixing condition on the underlying stationary marked point process which can be checked for a number of Poissonbased models and, in particular, in the case of geostatistical marking. Our method needs a central limit theorem for �-mixing random fields which is proved by extending Bernstein’s blocking technique to non-cubic index sets and seems to be of interest in its own right. By large-scale model-based simulations the performance of our test is studied in dependence of the model parameters which determine the range of spatial correlations.