Lothar Heinrich

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The paper introduces a family of stationary random measures in R d generated by so-called germ-grain models. The germ-grain model is deened as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures deened by the sum of contributions of non-overlapping parts of the individual(More)
We consider m-dependent random elds of bounded random vectors (generated by independend random elds) and investigate the analyticity of the cumulant generating function of sums of these random vectors. Using the Kirkwood-Salsburg equations we derive upper bounds for the cumulant generating function and prove its analyticity in a neighbourhood of zero, where(More)
We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in R d. This result generalizes an earlier one proved by Paroux [Adv. for intersection points of motion-invariant Poisson line processes in R 2. Our proof is based on Hoeffd-ing's decomposition of U-statistics which seems to be(More)
We give a representation of the second-order factorial moment measure of the point process of nodes (vertices of cells) associated with a stationary Voronoi tessellation in R d. If the Voronoi tessellation is generated by a stationary Poisson process this representation formula yields the corresponding pair correlation function g V (r) which can be(More)
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 vertices of the typical Poisson-Voronoi polyhedron(More)
We consider Johnson-Mehl tessellations generated by stationary independently marked (not necessarily Poissonian) point processes in d-dimensional Euclidean space. We rst analyze the Palm distribution of the thinned point process which coincides with the family of nuclei of non-empty Johnson-Mehl cells. This yields quite a general scheme for the construction(More)
For a sequence T (1) ; T (2) ; : : : of piecewise monotonic C 2-transformations of the unit interval I onto itself, we prove exponential-mixing , an almost Markov property and other higher-order mixing properties. Furthermore, we obtain optimal rates of convergence in the central limit theorem and large deviation relations for the sequence I are of bounded(More)
We investigate a class of kernel estimators σ 2 n of the asymptotic variance σ 2 of a d–dimensional stationary point process Ψ = i≥1 δ X i which can be observed in a cubic sampling window W n = [−n, n] d. σ 2 is defined by the asymptotic relation Var(Ψ(W n)) ∼ σ 2 (2n) d (as n → ∞) and its existence is guaranteed whenever the corresponding reduced(More)