# IJPAM: Volume 24, No. 2 (2005)

**RANDOM PARTITIONING PROBLEMS INVOLVING**

POISSON POINT PROCESSES ON THE INTERVAL

POISSON POINT PROCESSES ON THE INTERVAL

Laboratoire de Physique Théorique et Modélisation

CNRS-UMR 8089 et Université de Cergy-Pontoise

2 Avenue Adolphe Chauvin, 95032, Cergy-Pontoise, FRANCE

e-mail: Thierry.Huillet@ptm.u-cergy.fr

**Abstract.**Suppose some random resource (energy, mass or space) is to be
shared at random between (possibly infinitely many) species (atoms or
fragments). Assume
and suppose the amount of
the individual share is necessarily bounded from above by . This random
partitioning model can naturally be identified with the study of infinitely
divisible random variables with Lévy measure concentrated on the interval Special emphasis is put on these special partitioning models in the
Poisson-Kingman class. The masses attached to the atoms of such partitions
are sorted in decreasing order. Considering nearest-neighbors spacings
yields a partition of unity which also deserves special interest. For such
partition models, various statistical questions are addressed among which:
correlation structure, cumulative energy of the first largest items,
partition function, threshold and covering statistics, weighted partition,
Rényi's, typical and size-biased fragments size. Several physical images
are supplied.

When the unbounded Lévy measure of is , the spacings partition has Griffiths-Engen-McCloskey or GEM distribution and follows Dickman distribution. The induced partition models have many remarkable peculiarities which are outlined.

The case with finitely many (Poisson) fragments in the partition law is also
briefly addressed. Here, the Lévy measure is bounded.

**Received: **April 7, 2005

**AMS Subject Classification: **60G57, 62E17, 60K99, 62E15, 62E20

**Key Words and Phrases: **random partitions, divisibility, Poisson point process on the interval

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2005

**Volume:** 24

**Issue:** 2