Ratios of Ordered Points of Point Processes with Regularly Varying Intensity Measures

TL;DR

This paper investigates the asymptotic behavior of ratios of ordered points in point processes with regularly varying intensity measures, linking to extremal processes and generalized Poisson-Dirichlet distributions.

Contribution

It provides a systematic analysis of ratios of ordered points in such processes, connecting to extremal properties and generalized distributions.

Findings

Characterization of limiting ratios of ordered points

Connection to negative binomial and Poisson-Dirichlet processes

Framework for large-trimming properties of extremal processes

Abstract

We study limiting properties of ratios of ordered points of point processes whose intensity measures have regularly varying tails, giving a systematic treatment which points the way to "large-trimming" properties of extremal processes and a variety of applications. Our point process approach facilitates a connection with the negative binomial process of Gregoire (1984) and consequently to certain generalised versions of the Poisson-Dirichlet distribution.