### Samenvatting

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.
Keywords: Local times; Lévy processes; Infinite variance; Weak convergence; Crump–Mode–Jagers branching processes; Processor-Sharing queue

Originele taal-2 | Engels |
---|---|

Pagina's (van-tot) | 41-91 |

Aantal pagina's | 51 |

Tijdschrift | Journal of Theoretical Probability |

Volume | 28 |

Nummer van het tijdschrift | 1 |

DOI's | |

Status | Gepubliceerd - 2015 |

## Vingerafdruk Duik in de onderzoeksthema's van 'Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case'. Samen vormen ze een unieke vingerafdruk.

## Citeer dit

Lambert, A., & Simatos, F. (2015). Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case.

*Journal of Theoretical Probability*,*28*(1), 41-91. https://doi.org/10.1007/s10959-013-0492-1