Random Partitioning Problems Involving Poisson Point Processes On The Interval

TL;DR

This paper studies random partitioning models derived from Poisson point processes, focusing on their statistical properties, special cases like GEM distributions, and connections to infinitely divisible distributions with Lévy measures on the interval.

Contribution

It characterizes a class of partition models based on Poisson-Kingman processes, analyzing their statistical structure and special cases like the GEM distribution and Dickman distribution.

Findings

Analysis of correlation structures in partition models

Explicit formulas for the energy of largest fragments

Characterization of partition functions and size-biased distributions

Abstract

Suppose some random resource (energy, mass or space) $\chi \geq 0$ is to be shared at random between (possibly infinitely many) species (atoms or fragments). Assume ${\Bbb E}\chi =\theta <\infty $ and suppose the amount of the individual share is necessarily bounded from above by 1. This random partitioning model can naturally be identified with the study of infinitely divisible random variables with L\'{e}vy 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 $K$ largest items, partition function, threshold and covering statistics, weighted partition, R\'{e}nyi's, typical and size-biased fragments size. Several physical images are supplied. When the unbounded L\'{e}vy measure of $\chi $ is $\theta x^{-1}\cdot {\bf I}% (x\in (0,1)) dx$, the spacings partition has Griffiths-Engen-McCloskey or GEM$(\theta) $ distribution and $% \chi $ 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 L\'{e}vy measure is bounded.