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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 distributed. We are able to extend this central limit theorem to the total s-volume of the s-flats of all these polytopes contained in the expanding convex window, where s runs between 1 and d-1. It is noteworthy that the variances of these statistics are shown to be asymptotically proportional to some motion-invariant, non-decreasing, non-additive ovoid functional of the convex sampling body. Estimates and extremal properties of these functionals are of interest in convex geometry. Due to long-range dependences between distant parts of the generated Poisson hyperplane tessellation the variances increase faster than the volume of the sampling window. The proving technique used is based on Hoeffding's decomposition of U-statistics with Poisson distributed sample size. The obtained results generalize earlier ones proved in the special case of growing spherical windows.
For planar convex bodies K with positive area A(K), boundary length L(bd K) and second-order chord power integral I2(K), we study the ratio L(bd K)I2(K)/A(K)*A(K) and give reasons supporting the conjecture that its uniform lower bound is 32/3 attained exactly for circles. In particular, using the Ambartzumian-Pleijel representation of I2(K) we derive formulas for I2(K) in case of general triangles, rectangles, and regular N-gons. In these cases and for ellipses we can prove this inequality. A related conjecture is formulated for the class of convex N-gons which exhibits a strengthening of the isoperimetric inequality as well as of Carleman's inequality for convex N-gons. An extension to higher dimensions is discussed at the end of the paper.
A stationary Poisson cylinder process is composed by a stationary Poisson process of k-dimensional affine subspaces in a d-dimensional Euclidean space (1 <= k < d) which are dilated by a family of independent identically distributed random compact cylinder bases taken from the corresponding orthogonal complement. We study the accuracy of normal approximation of the d-volume of the union set of Poisson cylinders that covers cW when the scaling factor c > 0 becomes large. Here W denotes some fixed compact star-shaped window set containing the origin as inner point. We give lower and upper bounds of the variance of the union set in cW which exhibit long-range dependence within the union set of cylinders. Our main results are sharp estimates of the higher-order cumulants of the d-volume of all Poisson cylinders in cW 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' due to V. Statulevicius.
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 central limit theorems for the integrated squared error (ISE) of the empirical second-order product density and for the ISE of the empirical pair correlation function for expanding observation windows. The proof techniques are based on higher-order cumulant measures and the Brillinger-mixing property of the underlying point processes. The obtained Gaussian limits are used for constructing asymptotic goodness-of-fit tests for checking point process hypotheses even in the non-Poissonian case.
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 which is assumed to expand unboundedly in all directions. We first study the asymptotic behavior of the covariances of the empirical product densities under minimal moment and weak dependence assumptions. The proof of the main results is based on the Brillinger-mixing property of the underlying point process and certain smoothness conditions on the higher-order reduced cumulant measures. Finally, the obtained limit theorems allow to construct Chi-square-goodness-of-fit tests for hypothetical product densities.
We study some geometric inequalities for second-order chord power integrals I_2(K) of convex quadrangles K with positive area A(K) and boundary length L(bd K). Based on different representations of I_2(K) for convex quadrangles K we derive lower and upper bounds and give explicit formulas for I_2(K) in case of parallelograms and rhombs. Further, an elementary proof of the isoperimetric inequality and a Carleman-type inequality for quadrangles is given. At the end of the paper we state two conjectures on sharp upper and lower bounds of I_2(K) for convex polygons and their extensions to parallelotopes in higher dimensions.
Central Limit Theorem for the Volume of stationary Poisson Cylinder Processes in Expanding Domains
(2012)
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 typical cylinder base exists, we prove asymptotic normality of the d-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain r W as the scaling factor r grows unboundedly. Due to the long-range dependences within the union set of cylinders, the variance of its d-volume in r W increases asymptotically proportional to the (d+k)th power of r. To obtain the exact asymptotic behaviour of this variance we need a distinction between discrete and continuous directional distributions of the typical k-flat.
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-square-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 beta-mixing condition on the underlying stationary marked point process which can be checked for a number of Poisson-based models and, in particular, in the case of geostatistical marking. Our method needs a central limit theorem for beta-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.
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 beta-mixing) coefficient for point processes and derive an explicit condition in terms of this coefficient which implies finite total variation of the kth-order reduced factorial cumulant measure of the point process for fixed k >= 2. To prove this, we introduce higher-order covariance measures and use Statulevicius' representation formula for mixed cumulants in case of random (counting) measures. To illustrate our results, we consider some Brillinger-mixing point processes occurring in stochastic geometry.
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 quite different proofs. Further we determine the limit distributions of the maximal deviation and the integrated squared distance between empirical and true multiparameter second moment function. These results give rise to construct goodness-of-fit tests for checking the hypothesis that a given point pattern is completely spatially random (CSR), i.e. a realization of a homogeneous Poisson process.
First we discuss different representations of chord power integrals I_p(K) of any order p >= 0 for convex bodies K (with inner points) in the d-dimensional Euclidean space. Second we derive closed-term expressions of I_p(E(a)) for an ellipsoid E(a) with semi-axes a=(a_1,...,a_d) in terms of the support function of this ellipsoid and prove upper and lower bounds expressed by the volume and the mean breadth of E(a), respectively. A further inequality conjectured for convex body in Davy (1984) is proved for ellipsoids. Some remarks on chord power integrals of superellipsoids and simplices round off the topic. In the Appendix we prove a formula for the (d-1)-volume of (d-1)-ellipsoids arising from the intersection of E(a) with a hyperplane. Further, we derive the exact value of the third-order chord power integral of the Wuerfelecktetraeder correcting a wrong result by Emersleben (1962).
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 4 nearest neighbours. When τi denotes the even number of steps between the (i-1)-st and i-th return to the origin, we shall prove that the geometric mean of τ1,...,τn divided by npi converges in distribution to some positive random variable having a logarithmic stable law. We also obtain a rate of this convergence and improve an asymptotic estimate of the tail probability of τ1 due to Erdös and Taylor (1960).
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 it is shown that the normalized difference between scaled empirical and scaled theoretical K-function converges weakly to a mean zero Gaussian process with simple covariance function. This result suggests discrepancy measures between empirical and theoretical K-function with known limit distribution which allow to perform goodness-of-fit tests for checking a hypothesized point process based only on its intensity and (generalized) K-function. Similar test statistics are derived for testing the hypothesis that two independent point processes in W_n have the same distribution without explicit knowledge of their intensities and K-functions.
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 alpha-determinantal point process (DPP). Assuming absolute integrability of the function C_0(x) = C(o,x) we show that a stationary alpha-DPP with kernel function C_0(x) is "strongly" Brillinger-mixing implying, among others, that its tail-sigma-field is trivial. Second we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications to statistical second-order analysis of alpha-DPPs.
We consider a d-dimensional Boolean model Z = (Z_1+X_1) cup (Z_2+X_2) cup ... generated by a Poisson point process X_i, i = 1,2,... with some intensity measure and a sequence Z_i, i = 1,2,... of independent copies of some random compact set Z_0. Given compact sets K_1,...,K_l, we show that the discrete random vector (N(K_1),...,N(K_l)), where N(K_j) equals the number of shifted sets Z_i+X_i hitting K_j, obeys a l-variate Poisson distribution with 2^l-1 parameters. We obtain explicit formulae for all these parameters which can be estimated consistently from an observation of the union set Z in some unboundedly expanding window W_n (as n --> infty) provided that the Boolean model is stationary. Some of these results can be extended to unions of Poisson k-cylinders for k =1,...,d-1 and more general set-valued functionals of independently marked Poisson processes.
We prove some geometric inequalities for pth-order chord power integrals I_p(P_d), p=1,...,d, of d-parallelotopes P_d with positive volume V_d(P_d). First, we derive upper and lower bounds of the ratio I_p(P_d)/V_d^2(P_d) which are attained by a d-cuboid C_d with the same volume resp. the same mean breadth as P_d. Second, we apply the device of Schur-convexity to obtain bounds of I_p(C_d)/V_d^2(C_d) which are attained by a d-cube with the same volume resp. the same mean breadth as C_d. Most of these inequalities are shown for a more general class of ovoid functionals
containing, as by-product, a Pfiefer-type inequality for d-parallelotopes.
Suppose that a homogeneous system of spherical particles (d-spheres) with independent identically distributed radii is contained in some opaque d-dimensional body, and one is interested to estimate the common radius distribution. The only information one can get is by making a cross-section of that body with an s-flat (0 <= s <= d-1) and measuring the radii of the s-spheres and heir midpoints. The analytical solution of (the hyper-stereological version of) Wicksell's corpuscle problem is used to construct an empirical radius distribution of the d-spheres.
In this paper we study the asymptotic behaviour of this empirical radius distribution for s = d-1 and s = d-2 under the assumption that the intersection volume becomes unboundedly large and the point process of the midpoints of the d-spheres is Brillinger-mixing. Among others we generalize and extend some results obtained in [1] and [2] under the Poisson assumption for the special case d=3 and s=2.
When l probabilities are rounded to integer multiples of a given accuracy n, the sum of the numerators may deviate from n by a nonzero discrepancy. It is proved that, for large accuracies n --> infinty, the limiting discrepancy distribution has variance l/12. The relation to the uniform distribution over the interval [-1/2, 1/2], whose variance is 1/12, is explored in detail.
We study a particular class of stationary random closed sets in R^d called Poisson k-cylinder models (short: 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 corresponding P-k-CM is mixing. Regarding the P-(d-1)-CM as union of "thick hyperplanes" which generates a stationary process of polytopes we prove that the distribution of the polytope containing the origin does not depend on the thickness of the hyperplanes.
