Distances Between Poisson k-Flats

Article

Abstract

The distances between flats of a Poisson k-flat process in the d-dimensional Euclidean space with k < d/2 are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a given threshold and midpoint in a fixed compact and convex set is considered. For a family of increasing convex subsets, the asymptotic variance is computed and a central limit theorem with an explicit rate of convergence is proven. Moreover, the asymptotic distribution of the m-th smallest distance between two flats is investigated and it is shown that the ordered distances form asymptotically after suitable rescaling an inhomogeneous Poisson point process on the positive real half-axis. A similar result with a homogeneous limiting process is derived for distances around a fixed, strictly positive value. Our proofs rely on recent findings based on the Wiener–Itô chaos decomposition and the Malliavin–Stein method.

Keywords

Central limit theorem Chaos decomposition Extreme values Limit theorems Poisson flat process Poisson point process Poisson U-statistic Stochastic geometry Wiener–Itô integral 

AMS 2000 Subject Classification

60D05 60F05 60G55 60H07 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baumstark V, Last G (2009) Gamma distributions for stationary Poisson flat processes. Adv Appl Probab 41:911–939CrossRefMATHMathSciNetGoogle Scholar
  2. Bonnesen T, Fenchel W (1934) Theorie der konvexen Körper. Springer, BerlinCrossRefMATHGoogle Scholar
  3. Hug D, Last G, Weil W (2003) Distance measurements on processes of flats. Adv Appl Probab 35:70–95CrossRefMATHMathSciNetGoogle Scholar
  4. Hug D, Schneider R, Schuster R (2008) Integral geometry of tensor valuations. Adv Appl Math 41:482–509CrossRefMATHMathSciNetGoogle Scholar
  5. Kallenberg O (2002) Foundations of modern probability. Springer, New YorkCrossRefMATHGoogle Scholar
  6. Lachièze-Rey R, Peccati G (2012) Fine Gaussian fluctuations on the Poisson space I: contractions, cumulants and geometric random graphs. Electron J Probab, to appearGoogle Scholar
  7. Lachièze-Rey R, Peccati G (2012) Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometric U-statistics. arXiv:1205.0632
  8. Last G, Penrose MD (2011) Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab Theory Relat Fields 150:663–690CrossRefMATHMathSciNetGoogle Scholar
  9. Last G, Penrose MD, Schulte M, Thäle C (2012) Moments and central limit theorems for some multivariate Poisson functionals. arXiv:1205.3033
  10. Materon G (1975) Random sets and integral geometry. Wiley, New YorkGoogle Scholar
  11. Mecke J (1991) On the intersection density of flat processes. Math Nachr 151:69–74CrossRefMATHMathSciNetGoogle Scholar
  12. Mecke J, Thomas C (1986) On an extreme value problem for flat processes. Commun StatStoch Models 2:273–280CrossRefMATHMathSciNetGoogle Scholar
  13. Peccati G (2011) The Chen-Stein method for Poisson functionals. arXiv:1112.5051
  14. Peccati G, Taqqu MS (2008) Central limit theorems for double Poisson integrals. Bernoulli 14:791–821CrossRefMATHMathSciNetGoogle Scholar
  15. Peccati G, Solé JL, Taqqu MS, Utzet F (2010) Stein’s method and normal approximation of Poisson functionals. Ann Probab 38:443–478CrossRefMATHMathSciNetGoogle Scholar
  16. Reitzner M, Schulte M (2012) Central limit theorems for U-statistics of Poisson point processes. Ann Probab, to appearGoogle Scholar
  17. Schneider R (1999) A duality for Poisson flats. Adv Appl Probab 31:63–68CrossRefMATHGoogle Scholar
  18. Schneider R, Weil W (2008) Stochastic and integral geometry. Springer, BerlinCrossRefMATHGoogle Scholar
  19. Schulte M (2012) Normal approximation of Poisson functionals in Kolmogorov distance. arXiv:1206.3967
  20. Schulte M, Thäle C (2012) The scaling limit of Poisson-driven order statistics with applications in geometric probability. Stoch Process Their Appl 122:4096–4120CrossRefMATHGoogle Scholar
  21. Spodarev E (2001) On the rose of intersections of stationary flat processes. Adv Appl Probab 33:584–599CrossRefMATHMathSciNetGoogle Scholar
  22. Spodarev E (2003) Isoperimetric problems and roses of neighborhood for stationary flat processes. Math Nachr 251:88–100CrossRefMATHMathSciNetGoogle Scholar
  23. Weil W (1987) Point processes of cylinders, particles and flats. Acta Appl Math 9:103–136CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Fachbereich Mathematik/InformatikUniversität OsnabrückOsnabrückGermany
  2. 2.Ruhr University BochumBochumGermany

Personalised recommendations