Ratios of ordered points of point processes with regularly varying intensity measures

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.

Trimmed Lévy processes and their extremal components

“…One motivation for studying joint limit theorems as in Section 2 is to get information on limiting behaviour of ratios of a subordinator to its large jumps. See Ipsen et al (2018) and their references for related results and applications along these lines.…”

“….}. The trimmed subordinator is defined to be (r) X t = X t − r i=1 ∆ (i) t , t > 0, r ∈ N. In Buchmann et al (2018); Ipsen et al (2018) we considered distributional properties of ∆ (r) t as a function of r and here we continue that study by considering the joint weak limiting behaviour of (r) X t , ∆ (r) t as r → ∞. As r → ∞, (r) X t ↓ 0 and ∆ (r) t ↓ 0 a.s. for each t > 0, but conditionally on ∆ (r) t we may consider (r) X t as a Lévy process with Lévy measure restricted to (0, ∆ (r) t ) (e.g., Resnick (1986)).…”