The Annals of Applied Probability

Poisson approximation in connection with clustering of random points

Marianne Månsson

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 9, Number 2 (1999), 465-492.

Dates
First available in Project Euclid: 21 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1029962751

Digital Object Identifier
doi:10.1214/aoap/1029962751

Mathematical Reviews number (MathSciNet)
MR1687402

Zentralblatt MATH identifier
0941.60027

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 60G55: Point processes

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

Citation

Månsson, Marianne. Poisson approximation in connection with clustering of random points. Ann. Appl. Probab. 9 (1999), no. 2, 465--492. doi:10.1214/aoap/1029962751. https://projecteuclid.org/euclid.aoap/1029962751


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