2012/06/17 by Amaury Lambert, Lambert, Amaury, Florian Simatos +1
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.1206.3800
openalex publication_date 2012/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.