Central Limit Theorem for the Volume of stationary Poisson Cylinder Processes in Expanding Domains (original) (raw)
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Lithuanian Mathematical Journal, 2009
A stationary Poisson cylinder process Π (d,k) cyl 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 (d,k) of the union set of Π (d,k) cyl that covers W as the scaling factor 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 (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 (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.
Mixing properties of stationary Poisson cylinder models
Stochastics
We study a particular class of stationary random closed sets in R d called Poisson kcylinder 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.
On the central limit theorem for the stationary Poisson process of compact sets
Mathematische Nachrichten, 2004
Key words Central limit theorem, Poisson point process, random measure, segment process, space of compact sets MSC (2000) Primary: 60D05; Secondary: 60F05, 60G57 Stochastic geometry models based on a stationary Poisson point process of compact subsets of the Euclidean space are examined. Random measures on R 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.
On the Variance of the Area of Planar Cylinder Processes Driven by Brillinger-Mixing Point Processes
2021
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 processes form an important class of (in general non-stationary) planar random sets. We observe the cylinder process in an unboundedly growing domain ̺K when ̺ → ∞ , where the set K is compact and star-shaped w.r.t. the origin o being an inner point of K. Provided the unmarked point process satisfies a Brillinger-type mixing condition and the thickness of the typical cylinder has a finite second moment we prove a (weak) law of large numbers as well as a formula of the asymptotic variance for the area of the cylinder process in ̺K. Due to the long-range dependencies of the cylinder process, this variance increases proportionally to ̺.
Central limit theorems for Poisson hyperplane tessellations
The Annals of Applied Probability, 2006
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. in Appl. Probab. 30 (1998) 640-656] for intersection points of motioninvariant Poisson line processes in R 2 . Our proof is based on Hoeffding's decomposition of U -statistics which seems to be more efficient and adequate to tackle the higher-dimensional case than the "method of moments" used in [Adv. in Appl. Probab. 30 (1998) 640-656] to treat the case d = 2. Moreover, we extend our central limit theorem in several directions. First we consider k-flat processes induced by Poisson hyperplane processes in R d for 0 ≤ k ≤ d − 1. Second we derive (asymptotic) confidence intervals for the intensities of these k-flat processes and, third, we prove multivariate central limit theorems for the d-dimensional joint vectors of numbers of k-flats and their k-volumes, respectively, in an increasing spherical region.
Large deviations of the empirical volume fraction for stationary Poisson grain models
The Annals of Applied Probability, 2005
We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function Ln(z) = |Wn| −1 log E exp{z|Ξ ∩ Wn|} of the empirical volume fraction |Ξ ∩ Wn|/|Wn|, where | • | denotes the d-dimensional Lebesgue measure. Here Ξ = i≥1 (Ξi + Xi) denotes a d-dimensional Poisson grain model (also known as a Boolean model) defined by a stationary Poisson process Π λ = i≥1 δX i with intensity λ > 0 and a sequence of independent copies Ξ1, Ξ2,. .. of a random compact set Ξ0. For an increasing family of compact convex sets {Wn, n ≥ 1} which expand unboundedly in all directions, we prove the existence and analyticity of the limit limn→∞ Ln(z) on some disk in the complex plane whenever E exp{a|Ξ0|} < ∞ for some a > 0. Moreover, closely connected with this result, we obtain exponential inequalities and the exact asymptotics for the large deviation probabilities of the empirical volume fraction in the sense of Cramér and Chernoff.
Limit Theorems for Functionals of Random Convex Hulls in a Unit Disk
Mathematics and Statistics, 2023
In this article, we study the functionals of the convex hull generated by independent observations over two-dimensional random points. When the random points are given in the polar coordinate system, their components are independent of each other, the angular coordinate is distributed uniformly, and the tail of the distribution of the radial coordinate is a regularly varying function near the circle of the unit disk – support. Here, with the pproximation of the binomial point process by an inhomogeneous Poisson one, it is possible to study the asymptotic properties of the main functionals of the convex hull. Using the independence property of the increment of Poisson processes, we find an asymptotic expression for the mean values and variances for the main functionals of the convex hull. Uniform boundedness of exponential moments is proved for the same functionals, in the case when the convex hull is generated from an inhomogeneous Poisson point process inside the disk. The indicated independence property of the increment of the Poisson process allows us to express the area of the convex hull as a sum of independent identically distributed random variables, with which we prove the central limiting theorem for the number of vertices and the area of the convex hull. From the results obtained, we can conclude that if the tail of the distribution near the boundary is heavier, then there are many elements of the sample near the support boundary, and this means that there are many vertices of the convex hull, but the area bounded by the perimeter of the convex hull and the circle, as well as the difference between the perimeter of the convex hull and the circle, becomes negligible.
Lithuanian Mathematical Journal
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 of the union of strips contained in a space-filling window ϱK as ϱ → ∞. In the case the unmarked point process is Brillinger mixing, we prove themean-square convergence of the area fraction of the cylinder process in ϱK. Under stronger versions of Brillinger mixing, we obtain the exact variance asymptotics of the area of the cylinder process in ϱK as ϱ → ∞. Due to the long-range dependence of the cylinder process, this variance increases asymptotically proportionally to ϱ3.
Limit theorems for functionals on the facets of stationary random tessellations
Bernoulli, 2007
We observe stationary random tessellations X = {Ξn} n≥1 in 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 ≤ k ≤ d − 1) by their intersections with the (d − 1)-facets of independent and identically distributed motioninvariant tessellations Xn generated within each cell Ξn of X. The cases of X being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In both cases, however, we obtain that all of the total volumes measured in W are approximately normally distributed when W is sufficiently large. Structural formulae for mean values and asymptotic variances are derived and explicit numerical values are given for planar Poisson-Voronoi tessellations (PVTs) and Poisson line tessellations (PLTs).