Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The weak convergence of regenerative processes using some excursion path decompositions

Amaury Lambert and Florian Simatos

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Abstract

We consider regenerative processes with values in some general Polish space. We define their $\varepsilon$-big excursions as excursions $e$ such that $\varphi(e)>\varepsilon$, 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.

Résumé

Nous considérons des processus régénératifs à valeurs dans un espace polonais quelconque. Nous définissons leurs excursions $\varepsilon$-grandes comme les excursions $e$ telles que $\varphi(e)>\varepsilon$, où $\varphi$ est une fonctionnelle donnée sur l’espace des excursions, qui peut par exemple être la longueur ou la hauteur de $e$. Nous établissons une condition générale garantissant la convergence d’une suite de processus régénératifs, qui porte sur la convergence des excursions $\varepsilon$-grandes et de leurs extrémités, pour tout $\varepsilon$ dans un ensemble dont l’adhérence contient $0$. Enfin, nous donnons plusieurs conditions suffisantes sur les mesures d’excursion de cette suite pour que cette condition générale soit satisfaite, et nous discutons de possibles généralisations de notre approche à certains processus pouvant être écrits comme la concaténation de motifs i.i.d.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 492-511.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1395856137

Digital Object Identifier
doi:10.1214/12-AIHP531

Mathematical Reviews number (MathSciNet)
MR3189081

Zentralblatt MATH identifier
1291.60179

Subjects
Primary: 60K05: Renewal theory
Secondary: 60F05: Central limit and other weak theorems 60G07: General theory of processes 60G55: Point processes 60J55: Local time and additive functionals 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx]

Keywords
Regenerative process Excursion theory Excursion measure Weak convergence Queueing theory

Citation

Lambert, Amaury; Simatos, Florian. The weak convergence of regenerative processes using some excursion path decompositions. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 492--511. doi:10.1214/12-AIHP531. https://projecteuclid.org/euclid.aihp/1395856137


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