Lothar Heinrich

• University of Augsburg

We introduce cylinder processes in the plane defined as union sets of dilated straight lines (appearing as mutually overlapping infinitely long strips) generated by a stationary independently marked point process on the real line, where the marks describe the width and orientation of the individual cylinders. We study the behavior of the total area...

We study some asymptotic properties of cylinder processes in the plane defined as union sets of dilated straight lines (appearing as mutually overlapping infinitely long strips) derived from a stationary independently marked point process on the real line, where the marks describe thickness and orientation of individual cylinders. Such cylinder pro...

Recently, it has been proved that a stationary Brillinger-mixing point process is mixing (of any order) if its moment measures determine the distribution uniquely. In this paper we construct a family of non-ergodic stationary point processes as mixture of two distinct Brillinger-mixing Neyman–Scott processes having the same moment measures.

We study a particular class of stationary random closed sets in ℝd called Poisson k-cylinder models (P-k-CM’s) for k = 1,...,d - 1. We show that all P-k-CM’s are weakly mixing and possess long-range correlations. Further, we derive necessary and sufficient conditions in terms of the directional distribution of the cylinders under which the correspo...

We study sequences of scaled edge-corrected empirical (generalized) K-functions (modifying Ripley's K-function) each of them constructed from a single observation of a $d$-dimensional fourth-order stationary point process in a sampling window W_n which grows together with some scaling rate unboundedly as n --> infty. Under some natural assumptions...

When ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} probabilities are rounded to integer multiples of a given accuracy n, the sum of the numerat...

We prove some geometric inequalities for pth-order chord power integrals (Formula presented.) of d-parallelotopes (Formula presented.) with positive volume (Formula presented.). First, we derive upper and lower bounds of the ratio (Formula presented.) which are attained by a d-cuboid (Formula presented.) with the same volume resp. the same mean bre...

We consider a d-dimensional Boolean model \(\varXi = (\varXi _1+X_1)\cup (\varXi _2+X_2)\cup \cdots \) generated by a Poisson point process \(\{X_i, i\ge 1\}\) with intensity measure \(\varLambda \) and a sequence \(\{\varXi _i, i\ge 1\}\) of independent copies of some random compact set \(\varXi _0\,\). Given compact sets \(K_1,\ldots ,K_{\ell }\)...

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function C(x, y) defining an α-determinantal point process (DPP). Assuming absolute integrability of the function C 0(x) = C(o, x), we show that a stationary α-DPP with kernel function C 0(x) is “strongly” Brillinger-mixing, imp...

We prove two functional limit theorems for empirical multiparameter second moment functions (generalizing Ripley’s K-function) obtained from a homogeneous Poisson point field observed in an unboundedly expanding convex sampling window W n in ℝ d . The cases of known and unknown (estimated) intensity lead to distinct Gaussian limits and require qui...

First we discuss different representations of chord power integrals Ip(K) of any order p ≥ 0 for convex bodies K ⊂ R{double-struck}d with inner points. Second we derive closed-term expressions of Ip(E{double-struck}(a)) for an ellipsoid E{double-struck}(a) with semi-axes a = (a1,...,ad) in terms of the support function of E{double-struck}(a) and pr...

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 $\chi^2$-goodness-of-fit test. The corresponding test statistic is based on a natural empirical version of the Palm mark distribution and a s...

We consider the random walk of a particle on the two-dimensional integer lattice starting at the origin and moving from each site (independently of the previous moves) with equal probabilities to any of the four nearest neighbors. When τ i denotes the even number of steps between the (i − 1)th and ith returns to the origin, we shall prove that the...

We prove the asymptotic normality of kernel estimators of second- and higher-order product densities (and of the pair correlation function) for spatially homogeneous (and isotropic) point processes observed on a sampling window W n , which is assumed to expand unboundedly in all directions as n→∞. We first study the asymptotic behavior of the covar...

Marked point processes are stochastic models to describe random patterns of marked points {[X i ,M i ], i ≥ 1} in some bounded subset of the d-dimensional Euclidean space (usually d = 1, 2 or 3 in applications), where each point X i carries additional random information expressed as mark M i taking values in some metric space. To study the correlat...

We study the following problem: How to verify Brillinger-mixing of stationary point processes in ℝd by imposing conditions on a suitable mixing coefficient? For this, we define an absolute regularity (or β-mixing) coefficient for point processes and derive, in terms of this coefficient, an explicit condition that implies finite total variation of t...

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d-1) which are dilated by independent and identically distributed random compact cylinder bases taken from the corresponding (d-k)-dimensional orthogonal complement. If the second moment of the (d-k)-volume of t...

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 m...

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 \chi^2-goodness-of-fit test. The corresponding test statistic is based on a natural empirical version of the Palm mark distribution and a smo...

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed by a stationary Poisson process of k-flats (0 < k < d) which are dilated by independent identically distributed random compact cylinder bases taken from the corresponding (d-k)-dimensional orthogonal complement. If the second moment of the (d-k)-volume of the typ...

Spatial point processes are mathematical models for irregular or random point patterns in the d-dimensional space, where usually d = 2 or d = 3 in applications. The second-order product density and its isotropic analogue, the pair correlation function, are important tools for analyzing stationary point processes. In the present work we derive centr...

We investigate a class of kernel estimators \(\widehat{\sigma}^2_n\) of the asymptotic variance σ 2 of a d-dimensional stationary point process \(\Psi = \sum_{i\ge 1}\delta_{X_i}\) which can be observed in a cubic sampling window \(W_n = [-n,n]^d\,\). σ 2 is defined by the asymptotic relation \(Var(\Psi(W_n)) \sim \sigma^2 \,(2n)^d\) (as n → ∞) and...

We consider Wicksell's corpuscle pi'oblem for a homogeneous Poisson-grain model of (d + 1)' spheres in Rd+I, where only a single observation of the union set of partially or wholly occluded d-spheres in a sampling window W,, = 10, nfd contained in the d'dimensional intersection hyperplane is available. In contrast to the existing vast literature we...

A stationary Poisson cylinder process Π cyl (d,k) is composed of a stationary Poisson process of k-flats in ℝ d that are dilated by i.i.d. random compact cylinder bases taken from the corresponding orthogonal complement. We study the accuracy of normal approximation of the d-volume V ϱ (d,k) of the union set of Π cyl (d,k) that covers ϱW as the s...

For a sequence T(1), T(2),…of piecewise monotonic C2 - transformations of the unit interval I onto itself, we prove exponential ψ- 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 fk oT(k−1)o…oT(...

For planar convex bodies K with positive area A(K), boundary length L(K) and second-order chord power integral I_2(K), we study the ration L(K)I_2(K)/A^2(K) and give rasons supporting the conjecture that its uniform lower bound is 32/3 attained exactly for circles.In particular, using the Ambartzumian-Pleijel representation of I_2(K) we derive form...

In this paper we derive representation formulae for the second factorial moment measure of the point process of nodes and the second moment of the number of vertices of the typical cell associated with a stationary normal Voronoi tessellation in ℝd. In case the Voronoi tessellation is generated by a stationary Poisson process with intensity λ > 0 t...

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...

We consider motion-invariant (i.e. stationary and isotropic) Poisson hyperplane processes subdividing the d-dimensional Euclidean space into a collection of convex d-polytopes. Among others we prove that the total number of vertices of these polytopes lying in an unboundedly growing convex body (with inner points) is asymptotically normally distrib...

The paper ‘Modern statistics for spatial point processes’ by Jesper Møller and Rasmus P. Waagepetersen is based on a special invited lecture given by the authors at the 21st Nordic Conference on Mathematical Statistics, held at Rebild, Denmark, in June 2006. At the conference, Antti Penttinen and Eva B. Vedel Jensen were invited to discuss the pape...

We observe stationary random tessellations $X=\{\Xi_n\}_{n\ge1}$ in $\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1$) by their intersections with the $(d-1)$-facets of inde...

We consider a not necessarily stationary one-dimensional Boolean model Ξ=∪ i≥1(Ξ i +X i ) defined by a Poisson process Y = åi ³ 1dXi\Psi=\sum_{i\ge 1}\delta_{X_{i}} with bounded intensity function λ(t)≤λ 0 and a sequence of independent copies Ξ 1,Ξ 2,… of a random compact subset Ξ 0 of the real line ℝ1 whose diameter ‖Ξ 0‖ possesses a finite ex...

We study the limiting behaviour of suitably normalized union shot-noise processes , where F is a set-valued function on Rd × ℳ is a sequence of i.i.d. random elements on some measurable space [ℳ ] and Ψ = {xi, i≥ 1} stands for a stationary d-dimensional point process whose intensity λ tends to infinity. General results concerning weak convergence...

We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in $\mathbb{R}^d$. This result generalizes an earlier one proved by Paroux [Adv. in Appl. Probab. 30 (1998) 640--656] for intersection points of motion-invariant Poisson line processes in $\mathbb{R}^2$. Our proof is b...

Multiplier methods are used to round probabilities on finitely many categories to rational proportions. Focusing on the classical methods of Adams and Jefferson, we investigate goodness-of-fit criteria for this rounding process. Assuming that the given probabilities are uniformly distributed, we derive the limiting laws of the criteria, first when...

Germ-grain models are random closed sets in the d-dimensional Euclidean space ℝd which admit a representation as union of random compact sets (called grains) shifted by the atoms (called germs) of a point process. In this note we consider the distribution function F of an m-dimensional random vector describing shape and size parameters of the typic...

Nonparametric testing of distribution functions in germ-grain models / Z. Pawlas, L. Heinrich. - In: Case studies in spatial point process modeling / Adrian Baddeley ... (ed.). - New York : Springer, 2006. - S. 125-133. - (Lecture notes in statistics ; 185)

We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function L_n(z)=|W_n|^{-1}\logE\exp{z|\Xi\cap W_n|} of the empirical volume fraction |\Xi\cap W_n|/|W_n|, where |\cdot| denotes the d-dimensional Lebesgue measure. Here \Xi=\bigcup_{i\ge1}(\Xi_i+X_i) denotes a d-dimensional Poisson grain model (also known as a Boo...

Stationary multiplier methods are procedures for rounding real probabilities into rational proportions, while the Sainte-Laguë divergence is a reasonable measure for the cumulative error resulting from this rounding step. Assuming the given probabilities to be uniformly distributed, we show that the Sainte-Laguë divergences converge to the Lévy-sta...

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 independentl...

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. T...

In the present paper, we show how a consistent estimator can be derived for the asymptotic covariance matrix of stationary 0–1-valued vector fields in Rd, whose supports are jointly stationary random closed sets. As an example, which is of particular interest for statistical applications, we consider jointly stationary random closed sets associated...

Summary A non-parametric kernel estimator of the spectral density of stationary random closed sets is studied. Conditions are derived under which this estimator is asymptotically unbiased and mean-square consistent. For the planar Boolean model with isotropic compact and convex grains, an averaged version of the kernel estimator is compared with th...

For rounding arbitrary probabilities on finitely many categories to rational proportions, the multiplier method with standard rounding stands out. Sainte-Laguö showed in 1910 that the method minimizes a goodness-of-fit criterion that nowadays classifies as a chi-square divergence. Assuming the given probabilities to be uniformly distributed, we der...

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 ke...

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝ d generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. Th...

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in R^d generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual...

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in R d in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed...

We consider Johnson-Mehl tessellations generated by stationary independently marked (not necessarily Poissonian ) point processes in d-dimensional Euclidean space. We first 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...

. We consider m-dependent random fields of bounded random vectors (generated by independend random fields) 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...

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝ d in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed...

Consistency and asymptotic normality of a kernel-type estimator for the second-order product density of the stationary point process of exposed tangent points associated with a Boolean model are proved. This estimator is used to estimate characteristics of the typical grain. In the case of spherical grain we obtain an empirical nonparametric distri...

On asymptotic estimation in Wicksells corpuscle problem. - In: Symposium on Asymptotic Statistics <6, 1998, Praha>: Prague stochastics ´98 / ed.: Marie Husková ... - Prague : Union of Czech Mathematicians and Physicists. - Vol. 1. (1998). - S. 227-231

We determine the exact rate of Poisson approximation and give a second-order Poisson-Charlier expansion for the number of excedances of a given levelL n among the firstn digits of inhomogeneousf-expansions of real numbers in the unit interval. The application of this general result to homogeneousf-expansions and, in particular, to regular continue...

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. 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...

We introduce a family of stationary random measures in the Euclidean space generated by so-called germ-grain models. The germ-grain model is defined 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 defined by the sum of contributions of non-overlapping parts o...

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 Rd. If the Voronoi tessellation is generated by a stationary Poisson process this representation formula yields the corresponding pair correlation function gV (r) which can be e...

We give a bound for the absolute regularity coefficient of a Voronoi tessellation in terms of the absolute regularity coefficient of the generating stationary point process and its void probabilities. This result together with a suitable CLT for random fields, for example, provide asymptotic normality of the number of nodes or of the edges total le...

Using Hoeffding's decomposition and truncation arguments we prove some theorems on the conxergence of the distribution of suitably normalized U-statistics to an [alpha]-stable limit distribution, 1 < [alpha] <= 2, under very weak moment conditions on the kernel function.

Summary This paper presents a method for the estimation of parameters of random closed sets (racs’s) in ℝ d 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...

We give a rigorous definition of germ-grain models (ggm's) 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 g...

We give an upper bound for the absolute regularity coefficient of a stationary renewal process in terms of the total variation of the difference between the corresponding Palm and the usual renewal measure.

This paper suggests several tests for checking the nullhypothesis that a point pattern observed in a (large) rectangular sampling window DRd belongs to a realization of a stationary Poisson Process based on test statistics measuring the distance between the empirical and the true second moment function volume of the unit sphere im R d. We consider...

The main purpose of this paper is to present bounds of the discrepancy between an infinitely divisible distribution function with finite second moment and the normal as well as the poisson distribution function. Special emphasise is put on explicit numerical constancts involved in the error bounds. To improve these estimates an EDGEWORTH expansion...

For stationary POISSON cluster processes (PCP's) Ø on R the limit behaviour, as v(D) → ∞, of the quantity , where χ(x, r) = 1, if Ø(b(x, r)) = 1, and χ(x, r) = 0 otherwise, is studied. A central limit theorem for fixed r > 0 and the weak convergence of the normalized and centred empirical process on [0, R] to a continuous GAUSSian process are prove...

The main purpose of this paper is to present cer~trai limit tileoreins including functional limit theorems for empirical factorial moment measures and kernel-type product density estimators when the underlying point process is a regular infinitely divisible one.The requied moment conditions are minimal, they are necessary to ensure finite variances...

For a class of dependent random variables containing (i) Markov-dependent random variables with positive ergodicity coefficient, (ii) RMT-mixing and (iii) m-dependent random variablesa unified method for studying the limiting behaviour of sums is developed. This approach is used in various situations to derive rates of normal and stable convergence...

In the paper a sequence of bounded regions containing n independent identically and uniformly on Dn distributed points is considered. It is assumed that the d–dimensional volume v(Dn) is asymptotically proportional to n. Under these conditions it is shown that the number of pairs of points within a distance r>0 of each other is asymptotically norma...

Using a representation formula expressing the mixed cumulants of real-valued 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 ∞ ....

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 ∞. F...

In this note for i.i.d. random variables an equivalent formulation of SPITZER'S identity is given. This new form is generalized to the case of cyclically exchangeable random variables and proved by elementary probabilistic arguments. SPITZER'S identity itself does not hold for cyclically exchangeable random variables