Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case

TL;DR

This paper investigates the asymptotic behavior of local times of compound Poisson processes with drift and infinite variance, revealing convergence properties and implications for branching processes and queueing systems.

Contribution

It provides new results on the convergence of local times for compound Poisson processes with infinite variance, including weak convergence under specific jump distribution assumptions.

Findings

Finite-dimensional distributions converge under general assumptions.

Weak convergence achieved with additional jump distribution assumptions.

Results apply to stable Lévy processes with drift, impacting queueing theory and branching processes.

Abstract

Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive L\'evy process with non-zero L\'evy measure. In this paper we study the asymptotic behavior of the local time process, in the spatial variable, of these processes killed at two different random times: either at the time of the first visit of the L\'evy process to 0, in which case we prove results at the excursion level under suitable conditionings; or at the time when the local time at 0 exceeds some fixed level. We prove that finite-dimensional distributions converge under general assumptions, even if the limiting process is not c\`adl\`ag. Making an assumption on the distribution of the jumps of the compound Poisson processes, we strengthen this to get weak convergence. Our assumption allows for the limiting process to be a stable L\'evy process with drift. These results have implications on branching processes and in queueing theory, namely, on the scaling limit of binary, homogeneous Crump-Mode-Jagers processes and on the scaling limit of the Processor-Sharing queue length process.