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

Simatos, Florian and Lambert, Amaury Asymptotic Behavior of Local Times of Compound Poisson Processes with Drift in the Infinite Variance Case. (2015) Journal of Theoretical Probability, 28 (1). 41-91. ISSN 0894-9840

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Official URL: http://doi.org/10.1007/s10959-013-0492-1

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 excur- sion 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 gen- eral 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.

Item Type:Article
HAL Id:hal-01890474
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:Other partners > Sorbonne Université (FRANCE)
Other partners > Eindhoven University of Technology - TU/e (NETHERLANDS)
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Deposited On:08 Oct 2018 14:07

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