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Poisson Voronoi Tessellation In High Dimensions

Alishahi, Kasra | 2008

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 40133 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Zohori Zangeneh, Bijan; Shahshahani, Mehrdad
  7. Abstract:
  8. This thesis is devoted to study some asymptotic behaviors of Poisson Voronoi tessellation in the Euclidean space as dimension of the space tends to infinity. First we use Blaschke-Petkantschin formula to prove that the variance of volume of the typical cell tends to zero exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume converges in distribution to the constant . Next we consider the linear contact distribution function of the Poisson Voronoi tessellation and compute the limit when dimension goes to infinity. As a byproduct, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit. Finally we investigate the radii of the smallest ball containing and the largest ball contained in the typical cell, and prove that as dimension increases to infinity, the ratio of these two values becomes deterministic and tends to 2. To this purpose we use the notion of VC-dimension from discrete and computational geometry and prove an auxiliary result on bounding the probability that a Poisson point processes hits a family of given shapes simultaneously.


  9. Keywords:
  10. Stochastic Geometry ; Random Tessellation ; Poisson-Voronoi Tessellation ; High Dimention Data ; Limit Theorems

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