### Abstract

Original language | English |
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Publisher | s.n. |

Number of pages | 39 |

Publication status | Published - 2012 |

### Publication series

Name | arXiv |
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Volume | 1206.3800 [math.PR] |

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### Cite this

*Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case*. (arXiv; Vol. 1206.3800 [math.PR]). s.n.

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*Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case*. arXiv, vol. 1206.3800 [math.PR], s.n.

**Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case.** / Lambert, A.; Simatos, F.

Research output: Book/Report › Report › Academic

TY - BOOK

T1 - Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case

AU - Lambert, A.

AU - Simatos, F.

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

M3 - Report

T3 - arXiv

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

PB - s.n.

ER -