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.
Publié le : 1999-05-14
Classification:
Poisson approximation,
Stein's method,
total variation distance,
integral geometry,
convex sets,
mixed areas,
Poisson process,
60D05,
52A22,
60G55
@article{1029962751,
author = {M\aa nsson, Marianne},
title = {Poisson approximation in connection with clustering of random
points},
journal = {Ann. Appl. Probab.},
volume = {9},
number = {1},
year = {1999},
pages = { 465-492},
language = {en},
url = {http://dml.mathdoc.fr/item/1029962751}
}
Månsson, Marianne. Poisson approximation in connection with clustering of random
points. Ann. Appl. Probab., Tome 9 (1999) no. 1, pp. 465-492. http://gdmltest.u-ga.fr/item/1029962751/