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Asymptotic Behavior of Local Times of Compound Poisson Processes with Drift in the Infinite Variance Case

Author

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  • Amaury Lambert

    (UMR 7599 CNRS, UPMC Univ Paris 06)

  • Florian Simatos

    (Eindhoven University of Technology)

Abstract

Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive Lévy process with nonzero 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 càdlàg. 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.

Suggested Citation

  • Amaury Lambert & Florian Simatos, 2015. "Asymptotic Behavior of Local Times of Compound Poisson Processes with Drift in the Infinite Variance Case," Journal of Theoretical Probability, Springer, vol. 28(1), pages 41-91, March.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:1:d:10.1007_s10959-013-0492-1
    DOI: 10.1007/s10959-013-0492-1
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    References listed on IDEAS

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    1. Helland, Inge S., 1978. "Continuity of a class of random time transformations," Stochastic Processes and their Applications, Elsevier, vol. 7(1), pages 79-99, March.
    2. Kang, Ju-Sung & Wee, In-Suk, 1997. "A note on the weak invariance principle for local times," Statistics & Probability Letters, Elsevier, vol. 32(2), pages 147-159, March.
    3. Andreas E. Kyprianou & Víctor Rivero & Renming Song, 2010. "Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem," Journal of Theoretical Probability, Springer, vol. 23(2), pages 547-564, June.
    4. Sagitov, Serik, 1995. "A key limit theorem for critical branching processes," Stochastic Processes and their Applications, Elsevier, vol. 56(1), pages 87-100, March.
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    Cited by:

    1. Mijatović, Aleksandar & Uribe Bravo, Gerónimo, 2022. "Limit theorems for local times and applications to SDEs with jumps," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 39-56.
    2. Bert Zwart, 2022. "Conjectures on symmetric queues in heavy traffic," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 369-371, April.

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