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

A. Lambert, F. Simatos

Research output: Book/ReportReportAcademic

2 Downloads (Pure)

Abstract

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.
Original languageEnglish
Publishers.n.
Number of pages39
Publication statusPublished - 2012

Publication series

NamearXiv
Volume1206.3800 [math.PR]

Fingerprint

Infinite Variance
Compound Poisson Process
Local Time
Asymptotic Behavior
Scaling Limit
Jump
Limiting
Converge
Processor Sharing
Queueing Theory
Excursion
Queue Length
Branching process
Weak Convergence
Conditioning
Exceed
Binary

Cite this

Lambert, A., & Simatos, F. (2012). Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case. (arXiv; Vol. 1206.3800 [math.PR]). s.n.
@book{8eae1df9b1a44213b998d01841ea860c,
title = "Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case",
abstract = "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.",
author = "A. Lambert and F. Simatos",
year = "2012",
language = "English",
series = "arXiv",
publisher = "s.n.",

}

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

s.n., 2012. 39 p. (arXiv; Vol. 1206.3800 [math.PR]).

Research output: Book/ReportReportAcademic

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 -