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

Abstract : Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive Lévy process with non-zero Lévy 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 Lévy 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 cádlág. 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 Lévy 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.
Type de document :
Pré-publication, Document de travail
39p. 2012
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https://hal.archives-ouvertes.fr/hal-00710037
Contributeur : Amaury Lambert <>
Soumis le : mardi 19 juin 2012 - 23:30:18
Dernière modification le : mardi 11 octobre 2016 - 14:34:17

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  • HAL Id : hal-00710037, version 1
  • ARXIV : 1206.3800

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UPMC | INSMI | PMA | USPC

Citation

Amaury Lambert, Florian Simatos. Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case. 39p. 2012. <hal-00710037>

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