## The Annals of Applied Probability

### Poisson approximation in connection with clustering of random points

Marianne Månsson

#### 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

https://projecteuclid.org/euclid.aoap/1029962751

Digital Object Identifier
doi:10.1214/aoap/1029962751

Mathematical Reviews number (MathSciNet)
MR1687402

Zentralblatt MATH identifier
0941.60027

#### 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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