The weak convergence of regenerative processes using some excursion path decompositions
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 492-511.

We consider regenerative processes with values in some general Polish space. We define their ε-big excursions as excursions e such that ϕ(e)>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.

Nous considérons des processus régénératifs à valeurs dans un espace polonais quelconque. Nous définissons leurs excursions ε-grandes comme les excursions e telles que ϕ(e)>ε, où ϕ 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 ε-grandes et de leurs extrémités, pour tout ε 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.

DOI: 10.1214/12-AIHP531
Classification: 60K05, 60F05, 60G07, 60G55, 60J55, 60K20
Keywords: regenerative process, excursion theory, excursion measure, weak convergence, queueing theory
@article{AIHPB_2014__50_2_492_0,
     author = {Lambert, Amaury and Simatos, Florian},
     title = {The weak convergence of regenerative processes using some excursion path decompositions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {492--511},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     doi = {10.1214/12-AIHP531},
     mrnumber = {3189081},
     zbl = {1291.60179},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/12-AIHP531/}
}
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
SP  - 492
EP  - 511
VL  - 50
IS  - 2
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/12-AIHP531/
DO  - 10.1214/12-AIHP531
LA  - en
ID  - AIHPB_2014__50_2_492_0
ER  - 
%0 Journal Article
%A Lambert, Amaury
%A Simatos, Florian
%T The weak convergence of regenerative processes using some excursion path decompositions
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 492-511
%V 50
%N 2
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/12-AIHP531/
%R 10.1214/12-AIHP531
%G en
%F AIHPB_2014__50_2_492_0
Lambert, Amaury; Simatos, Florian. The weak convergence of regenerative processes using some excursion path decompositions. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 492-511. doi : 10.1214/12-AIHP531. http://archive.numdam.org/articles/10.1214/12-AIHP531/

[1] D. Aldous. Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 (1997) 812-854. | MR | Zbl

[2] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York, 1999. | MR | Zbl

[3] R. M. Blumenthal. Excursions of Markov Processes. Probability and Its Applications. Birkhäuser Boston, Boston, MA, 1992. | MR | Zbl

[4] S. Borst and F. Simatos. A stochastic network with mobile users in heavy traffic. Queueing Systems Theory Appl. To appear. DOI:10.1007/s11134-012-9330-x. Available at arXiv:1202.2881. | MR | Zbl

[5] H. C. Gromoll. Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab. 14 (2004) 555-611. | MR | Zbl

[6] H. C. Gromoll, A. L. Puha and R. J. Williams. The fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab. 12 (2002) 797-859. | MR | Zbl

[7] K. Itô. Poisson point processes attached to Markov processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability Theory 225-239. Univ. California Press, Berkeley, CA, 1972. | MR | Zbl

[8] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer-Verlag, Berlin, 2003. | MR | Zbl

[9] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and Its Applications (New York). Springer-Verlag, New York, 2002. | MR | Zbl

[10] M. J. Kozdron. On the scaling limit of simple random walk excursion measure in the plane. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006) 125-155. | MR | Zbl

[11] A. Lambert and F. Simatos. Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case. Preprint, 2012, available at arXiv:1206.3800. | MR

[12] A. Lambert, F. Simatos and B. Zwart. Scaling limits via excursion theory: Interplay between Crump-Mode-Jagers branching processes and Processor-Sharing queues. Ann. Appl. Probab. To appear. Available at arXiv:1102.5620. | MR | Zbl

[13] G. F. Lawler and J. A. Trujillo Ferreras. Random walk loop soup. Trans. Amer. Math. Soc. 359 (2007) 767-787. | MR | Zbl

[14] E. Pardoux and A. Wakolbinger. From exploration paths to mass excursions - variations on a theme of Ray and Knight. In Surveys in Stochastic Processes 87-106. J. Blath, P. Imkeller and S. Roelly (Eds.). EMS Series of Congress Reports. Eur. Math. Soc., Zürich, 2011. | MR | Zbl

[15] C. Stone. Weak convergence of stochastic processes defined on semi-infinite time intervals. Proc. Amer. Math. Soc. 14 (1963) 694-696. | MR | Zbl

[16] W. Whitt. Heavy-traffic limits for the G/H 2 * /n/m queue. Math. Oper. Res. 30 (2005) 1-27. | MR | Zbl

Cited by Sources: