The weak convergence of regenerative processes using some excursion path decompositions

Amaury Lambert; Florian Simatos

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 2, page 492-511
  • ISSN: 0246-0203

Abstract

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We consider regenerative processes with values in some general Polish space. We define their ε -big excursions as excursions e such that ϕ ( e ) g t ; ε , where ϕ is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of e . We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of ε -big excursions and of their endpoints, for all ε in a set whose closure contains 0 . Finally, we provide various sufficient conditions on the excursion measures of this sequence for this general condition to hold and discuss possible generalizations of our approach to processes that can be written as the concatenation of i.i.d. motifs.

How to cite

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Lambert, Amaury, and Simatos, Florian. "The weak convergence of regenerative processes using some excursion path decompositions." Annales de l'I.H.P. Probabilités et statistiques 50.2 (2014): 492-511. <http://eudml.org/doc/272005>.

@article{Lambert2014,
abstract = {We consider regenerative processes with values in some general Polish space. We define their $\varepsilon $-big excursions as excursions $e$ such that $\varphi (e)&gt;\varepsilon $, where $\varphi $ is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of $e$. We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of $\varepsilon $-big excursions and of their endpoints, for all $\varepsilon $ in a set whose closure contains $0$. Finally, we provide various sufficient conditions on the excursion measures of this sequence for this general condition to hold and discuss possible generalizations of our approach to processes that can be written as the concatenation of i.i.d. motifs.},
author = {Lambert, Amaury, Simatos, Florian},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {regenerative process; excursion theory; excursion measure; weak convergence; queueing theory},
language = {eng},
number = {2},
pages = {492-511},
publisher = {Gauthier-Villars},
title = {The weak convergence of regenerative processes using some excursion path decompositions},
url = {http://eudml.org/doc/272005},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Lambert, Amaury
AU - Simatos, Florian
TI - The weak convergence of regenerative processes using some excursion path decompositions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 2
SP - 492
EP - 511
AB - We consider regenerative processes with values in some general Polish space. We define their $\varepsilon $-big excursions as excursions $e$ such that $\varphi (e)&gt;\varepsilon $, where $\varphi $ is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of $e$. We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of $\varepsilon $-big excursions and of their endpoints, for all $\varepsilon $ in a set whose closure contains $0$. Finally, we provide various sufficient conditions on the excursion measures of this sequence for this general condition to hold and discuss possible generalizations of our approach to processes that can be written as the concatenation of i.i.d. motifs.
LA - eng
KW - regenerative process; excursion theory; excursion measure; weak convergence; queueing theory
UR - http://eudml.org/doc/272005
ER -

References

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