Regenerative Compositions in the Case of Slow Variation

TL;DR

This paper investigates the asymptotic behavior of the number of gaps hit by a Poisson process in the range of a subordinator with slowly varying Lévy measure tail, extending previous studies.

Contribution

It extends prior work by analyzing fluctuations of the process counting hit gaps for subordinators with slowly varying Lévy measure tails.

Findings

Asymptotic fluctuation results for the process ${\\cal K}_n$ as $n\to\infty$

Generalization of previous models to slowly varying Lévy measures

Broad spectrum of situations analyzed

Abstract

For $S$ a subordinator and $\Pi_n$ an independent Poisson process of intensity $ne^{-x}, x>0,$ we are interested in the number $K_n$ of gaps in the range of $S$ that are hit by at least one point of $\Pi_n$. Extending previous studies in \cite{Bernoulli, GPYI, GPYII} we focus on the case when the tail of the L{\'e}vy measure of $S$ is slowly varying. We view $K_n$ as the terminal value of a random process ${\cal K}_n$, and provide an asymptotic analysis of the fluctuations of ${\cal K}_n$, as $n\to\infty$, for a wide spectrum of situations.