On the central limit theorem for the stationary Poisson process of compact sets

Stochastic geometry models based on a stationary Poisson point process of compact subsets of the Euclidean space are examined. Random measures on ?^d, derived from these processes using Hausdorff and projection measures are studied. The central limit theorem is formulated in a way which enables comparison of the various estimators of the intensity of the produced random measures. Approximate confidence intervals for the intensity are constructed. Their use is demonstrated in an example of length intensity estimation for the segment processes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second‐order properties of the point process of nodes in a stationary Voronoi tessellation

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 ?^d . In case the Voronoi tessellation is generated by a stationary Poisson process with intensity λ > 0 the corresponding pair correlation function g_V,λ (r) can be expressed by a weighted sum of d +2 (numerically tractable) multiple parameter integrals. The asymptotic variance of the number of nodes in an increasing cubic domain as well as the second moment of the number of vertices of the typical Poisson Voronoi cell are calculated exactly by means of these parameter integrals. The existence of a (d ? 1)st‐order pole of g_V,λ (r) at r = 0 is proved and the exact value of lim_{r →0} r^{d –1} g_V,λ (r) is determined. In the particular cases d = 2 and d = 3 the graph of g_V,1(r) including its local extreme points, the points of level 1 of gV, 1(r) and other characteristics are computed by numerical integration. Furthermore, an asymptotically exact confidence interval for the intensity of nodes is obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Some Inequalities for Poisson Networks

A stationary Poisson hyperplane process in R^d induces a random network of (d-2)-flats, each of which is the intersection of two hyperplanes of the process. It is known that the intensity of the induced (d-2)-flat process divided by the square of the intensity of the original hyperplane process is maximal in the isotropic case. An integral-geometric formula for elliptic spaces is presented, from which the mentioned extremum property and related inequalities for superpositions of stationary Poisson hyperplane processes are derived.     

Central limit theorems for the radial spanning tree

Consider a homogeneous Poisson point process in a compact convex set in d‐dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 262–286, 2017     

Percolation and limit theory for the poisson lilypond model

The lilypond model on a point process in d ‐space is a growth‐maximal system of non‐overlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a sequence of Poisson or binomial point processes on expanding windows. For the lilypond model over a homogeneous Poisson process, we give subexponentially decaying tail bounds for the size of the cluster at the origin. Finally, we consider the enhanced Poisson lilypond model where all the balls are enlarged by a fixed amount (the enhancement parameter), and show that for d > 1 the critical value of this parameter, above which the enhanced model percolates, is strictly positive. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Multiple Phase Transitions in Long‐Range First‐Passage Percolation on Square Lattices

We consider a model of long‐range first‐passage percolation on the d‐dimensional square lattice ?d in which any two distinct vertices x,y ? ?^d are connected by an edge having exponentially distributed passage time with mean ‖ x – y ‖^α+o(1), where α > 0 is a fixed parameter and ‖·‖ is the l¹–norm on ?^d. We analyze the asymptotic growth rate of the set ß_t, which consists of all x ? ?^d such that the first‐passage time between the origin 0 and x is at most t as t → ∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α < ^d, (ii) stretched exponential growth for α ? d,2d), (iii) superlinear growth for α ? (2d,2d + 1), and finally (iv) linear growth for α > 2d + 1 like the nearest‐neighbor first‐passage percolation model corresponding to α=∞. © 2015 Wiley Periodicals, Inc.     

Some Limit Theorems for Extremal and Union Shot-Noise Processes

We study the limiting behaviour of suitably normalized union shot-noise processes , where F is a set-valued function on R^d × ?? is a sequence of i.i.d. random elements on some measurable space [?? ??] and Ψ = {x_i, i≥ 1} stands for a stationary d-dimensional point process whose intensity λ tends to infinity. General results concerning weak convergence of parametrized union shot-noise processes Ξ?(t) as ? ↓ 0 are obtained (Theorem 5.1 and its corollaries), if the point process λ^{1 d}Ψ has a weak limit and F satisfies some technical conditions. An essential tool for proving these results is the notion of regular variation of multivalued functions. Some examples illustrate the applicability of the results.     

Variational Formula for the Time Constant of First‐Passage Percolation

We consider first‐passage percolation with positive, stationary‐ergodic weights on the square lattice ?^d. Let T(x) be the first‐passage time from the origin to a point x in ?^d. The convergence of the scaled first‐passage time T([nx])/n to the time constant as n → ∞ can be viewed as a problem of homogenization for a discrete Hamilton‐Jacobi‐Bellman (HJB) equation. We derive an exact variational formula for the time constant and construct an explicit iteration that produces a minimizer of the variational formula (under a symmetry assumption). We explicitly identify when the iteration produces correctors.© 2016 Wiley Periodicals, Inc.     

Local empirical processes near boundaries of convex bodies

We investigate the behaviour of Poisson point processes in the neighbourhood of the boundary ∂K of a convex body K in ,d ≥ 2. Making use of the geometry of K, we show various limit results as the intensity of the Poisson process increases and the neighbourhood shrinks to ∂K. As we shall see, the limit processes live on a cylinder generated by the normal bundle of K and have intensity measures expressed in terms of the support measures of K. We apply our limit results to a spatial version of the classical change-point problem, in which random point patterns are considered which have different distributions inside and outside a fixed, but unknown convex body K.     

Percolation in invariant Poisson graphs with?i.i.d.?degrees

Let each point of a homogeneous Poisson process in ℝ ^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution μ on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution μ. Leaving aside degenerate cases, we prove that for any μ there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on μ for the absence and presence of infinite components.     

Inequalities for Stationary Poisson Cuboid Processes

A cuboid is a rectangular parallelepipedon. By the notion “stationary Poisson cuboid process” we understand a stationary Poisson hyperplane process which divides the Euclidean space ?^d into cuboids. It is equivalent to speak of a stationary Poisson cuboid tessellation. The distributions of volume and total edge length of the typical cuboid and the origin-cuboid of a stationary Poisson cuboid process are considered. It is shown that these distributions become minimal, in the sense of a specific order relation, in the case of quasi-isotropy. A possible connection to a more general problem, treated in [6], is also discussed.     

Confinement of Brownian motion among Poissonian obstacles in ℝ d , d≥ 3

Consider Brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of ℝ ^d , d≥ 1, at the points of a Poisson cloud of constant intensity v <: 0. Assume that Brownian motion is absorbed instantaneously upon entering the obstacle set. In SZN-conf Sznitman has shown that in d = 2, conditionally on the event that the process does not enter the obstacle set up to time t, the probability that Brownian motion remains within distance ∼t ^1/4 from its starting point is going to 1 as t goes to infinity. We show that the same result holds true for d≥ 3, with t ^1/4 replaced by t ^{1/( d +2)}. The proof is based on Sznitmans refined method of enlargement of obstacles [10] as well as on a quantitative isoperimetric inequality due to Hall [4]. Received: 6 July 1998     

Berry–Esseen bounds and Cramér-type large deviations for the volume distribution of Poisson cylinder processes

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 scaling factor ϱ becomes large. Here W is some fixed compact star-shaped set containing the origin as an inner point. We give lower and upper bounds of the variance of V _ϱ ^(d,k) that exhibit long-range dependence within the union set of cylinders. Our main results are sharp estimates of the higher-order cumulants of V _ϱ ^(d,k) under the assumption that the (d − k)-volume of the typical cylinder base possesses a finite exponential moment. These estimates enable us to apply the celebrated “Lemma on large deviations” of Statulevičius.     

Limit theory for the random on‐line nearest‐neighbor graph

In the on‐line nearest‐neighbor graph (ONG), each point after the first in a sequence of points in ?^d is joined by an edge to its nearest neighbor amongst those points that precede it in the sequence. We study the large‐sample asymptotic behavior of the total power‐weighted length of the ONG on uniform random points in (0,1)^d. In particular, for d = 1 and weight exponent α > 1/2, the limiting distribution of the centered total weight is characterized by a distributional fixed‐point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest‐neighbor (directed) graph on uniform random points in the unit interval. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

On lattice points in polyhedral cross-sections

(a) We prove that the convex hull of anyk ^d +1 points of ad-dimensional lattice containsk+1 collinear lattice points. (b) For a convex polyhedron we consider the numbers of its lattice points in consecutive parallel lattice hyperplanes (levels). We prove that if a polyhedron spans no more than 2 ^d−1 levels, then this string of numbers may be arbitrary. On the other hand, we give an example of a string of 2 ^d−1+1 numbers to which no convex polyhedron corresponds inR ^d .     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.     

Structure of the Particle Population for a Branching Random Walk with a Critical Reproduction Law

We consider a continuous-time symmetric branching random walk on the d-dimensional lattice, d ≥?1, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk has a finite variance of jumps and the reproduction law is described by a continuous-time Markov branching process (a continuous-time analog of a Bienamye-Galton-Watson process) at every lattice point. We study the structure of the particle subpopulation generated by the initial particle situated at a lattice point x. We replay why vanishing of the majority of subpopulations does not affect the convergence to the steady state and leads to clusterization for lattice dimensions d =?1 and d =?2.

     

Nonparametric Estimation of Regression Functions in Point Process Models

We prove that the empirical L ²-risk minimizing estimator over some general type of sieve classes is universally, strongly consistent for the regression function in a class of point process models of Poissonian type (random sampling processes). The universal consistency result needs weak assumptions on the underlying distributions and regression functions. It applies in particular to neural net classes and to radial basis function nets. For the estimation of the intensity functions of a Poisson process a similar technique yields consistency of the sieved maximum likelihood estimator for some general sieve classes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Vertex Numbers of Weighted Faces in Poisson Hyperplane Mosaics

In the random mosaic generated by a stationary Poisson hyperplane process in ℝ ^d , we consider the typical k-face weighted by the j-dimensional volume of the j-skeleton (0≤j≤k≤d). We prove sharp lower and upper bounds for the expected number of its vertices.     

Small faces in stationary Poisson hyperplane tessellations

We consider the tessellation induced by a stationary Poisson hyperplane process in d‐dimensional Euclidean space. Under a suitable assumption on the directional distribution, and measuring the k‐faces of the tessellation by a suitable size functional, we determine a limit distribution for the shape of the typical k‐face, under the condition of given size and this tending to zero. The limit distribution is concentrated on simplices. This extends a result of Gilles Bonnet.