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May 1999 Poisson approximation in connection with clustering of random points
Marianne Månsson
Ann. Appl. Probab. 9(2): 465-492 (May 1999). DOI: 10.1214/aoap/1029962751


Let n particles be independently and uniformly distributed in a rectangle $\mathbf{A} \subset \mathbb{R}^2$. Each subset consisting of $k \leq n$ particles may possibly aggregate in such a way that it is covered by some translate of a given convex set $C \subset \mathbf{A}$. The number of k-subsets which actually are covered by translates of C is denoted by W. The positions of such subsets constitute a point process on A. Each point of this process can be marked with the smallest necessary "size" of a set, of the same shape and orientation as C, which covers the particles determining the point. This results in a marked point process.

The purpose of this paper is to consider Poisson process approximations of W and of the above point processes, by means of Stein's method. To this end, the exact probability for k specific particles to be covered by some translate of C is given.


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Marianne Månsson. "Poisson approximation in connection with clustering of random points." Ann. Appl. Probab. 9 (2) 465 - 492, May 1999.


Published: May 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0941.60027
MathSciNet: MR1687402
Digital Object Identifier: 10.1214/aoap/1029962751

Primary: 60D05
Secondary: 52A22 , 60G55

Keywords: convex sets , integral geometry , mixed areas , Poisson approximation , Poisson process , Stein's method , total variation distance

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.9 • No. 2 • May 1999
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