The scaling limits of a heavy tailed Markov renewal process

Julien Sohier

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 2, page 483-505
  • ISSN: 0246-0203

Abstract

top
In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the α -stable regenerative set. We then apply these results to the strip wetting model which is a random walk S constrained above a wall and rewarded or penalized when it hits the strip [ 0 , ∞ ) × [ 0 , a ] where a is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.

How to cite

top

Sohier, Julien. "The scaling limits of a heavy tailed Markov renewal process." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 483-505. <http://eudml.org/doc/271967>.

@article{Sohier2013,
abstract = {In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the $\alpha $-stable regenerative set. We then apply these results to the strip wetting model which is a random walk $S$ constrained above a wall and rewarded or penalized when it hits the strip $[0,\infty )\times [0,a]$ where $a$ is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.},
author = {Sohier, Julien},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Heavy tailed Markov renewals processes; scaling limits; fluctuation theory for random walks; regenerative sets; Heavy tailed Markov renewal process; scaling limit; fluctuation theory; random walk; regenerative set},
language = {eng},
number = {2},
pages = {483-505},
publisher = {Gauthier-Villars},
title = {The scaling limits of a heavy tailed Markov renewal process},
url = {http://eudml.org/doc/271967},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Sohier, Julien
TI - The scaling limits of a heavy tailed Markov renewal process
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 483
EP - 505
AB - In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the $\alpha $-stable regenerative set. We then apply these results to the strip wetting model which is a random walk $S$ constrained above a wall and rewarded or penalized when it hits the strip $[0,\infty )\times [0,a]$ where $a$ is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.
LA - eng
KW - Heavy tailed Markov renewals processes; scaling limits; fluctuation theory for random walks; regenerative sets; Heavy tailed Markov renewal process; scaling limit; fluctuation theory; random walk; regenerative set
UR - http://eudml.org/doc/271967
ER -

References

top
  1. [1] S. Asmussen. Applied Probability and Queues, 2nd edition. Applications of Stochastic Modelling and Applied Probability 51. Springer, New York, 2003. Zbl1029.60001MR1978607
  2. [2] J. Bertoin. Subordinators: Examples and applications. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997) 1–91. Lecture Notes in Math. 1717. Springer, Berlin, 1999. Zbl0955.60046MR1746300
  3. [3] A. N. Borodin and P. Salminen. Handbook of Brownian Motion – Facts and Formulae, 2nd edition. Probability and Its Applications. Birkhäuser, Basel, 2002. Zbl0859.60001MR1912205
  4. [4] F. Caravenna and J.-D. Deuschel. Pinning and wetting transition for ( 1 + 1 ) -dimensional fields with Laplacian interaction. Ann. Probab.36 (2008) 2388–2433. Zbl1179.60066MR2478687
  5. [5] F. Caravenna and J.-D. Deuschel. Scaling limits of ( 1 + 1 ) -dimensional pinning models with Laplacian interaction. Ann. Probab.37 (2009) 903–945. Zbl1185.60106MR2537545
  6. [6] F. Caravenna, G. Giacomin and L. Zambotti. Tightness conditions for polymer measures. Preprint, 2007. Available at arXiv.org:math/0702331. Zbl1138.60060
  7. [7] F. Caravenna, G. Giacomin and L. Zambotti. Sharp asymptotic behavior for wetting models in ( 1 + 1 ) -dimension. Electron. J. Probab. 11 (2006) 345–362 (electronic). Zbl1112.60068MR2217821
  8. [8] F. Caravenna, G. Giacomin and L. Zambotti. Infinite volume limits of polymer chains with periodic charges. Markov Process. Related Fields13 (2007) 697–730. Zbl1138.60060MR2381597
  9. [9] E. Çinlar. Some joint distributions for Markov renewal processes. Aust. N. Z. J. Stat.10 (1968) 8–20. Zbl0162.48701MR233432
  10. [10] M. Cranston, L. Koralov, S. Molchanov and B. Vainberg. Continuous model for homopolymers. J. Funct. Anal.256 (2009) 2656–2696. Zbl1162.82031MR2502529
  11. [11] J.-D. Deuschel, G. Giacomin and L. Zambotti. Scaling limits of equilibrium wetting models in ( 1 + 1 ) -dimension. Probab. Theory Related Fields132 (2005) 471–500. Zbl1084.60060MR2198199
  12. [12] R. A. Doney. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields107 (1997) 451–465. Zbl0883.60022MR1440141
  13. [13] R. A. Doney. Local behavior of first passage probabilities. Probab. Theory Related Fields5 (2010) 299–315. Zbl1237.60036MR2892956
  14. [14] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York, 1971. Zbl0138.10207MR270403
  15. [15] P. J. Fitzsimmons, B. Fristedt and B. Maisonneuve. Intersections and limits of regenerative sets. Z. Wahrsch. Verw. Gebiete70 (1985) 157–173. Zbl0548.60084MR799144
  16. [16] G. Giacomin. Random Polymer Models. Imperial College Press, London, 2007. Zbl1125.82001MR2380992
  17. [17] P. E. Greenwood and W. Wefelmeyer. Empirical estimators for semi-Markov processes. Math. Meth. Statist.5 (1996) 299–315. Zbl0872.62038MR1417674
  18. [18] P. Lévy. Processus semi-markoviens. In Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Vol. III 416–426. Erven P. Noordhoff N.V., Groningen, 1956. Zbl0073.34705MR88105
  19. [19] R. Pyke. Markov renewal processes with finitely many states. Ann. Math. Statist.32 (1961) 1243–1259. Zbl0201.49901MR154324
  20. [20] R. Pyke and R. Schaufele. Limit theorems for Markov renewal processes. Ann. Math. Statist.35 (1964) 1746–1764. Zbl0134.34602MR168026
  21. [21] W. L. Smith. Regenerative stochastic processes. Proc. R. Soc. Lond. Ser. A232 (1955) 6–31. Zbl0067.36301MR73877
  22. [22] J. Sohier. A functional limit convergence towards Brownian excursion. Preprint, 2010. Available at arXiv.org:1012.0118. Zbl1271.60095
  23. [23] J. Sohier. On pinning phenomena and random walk fluctuation theory. Ph.D. thesis, Univ. Paris 7, France, 2010. Available at http://hal.archives-ouvertes.fr/tel-00534716/. 
  24. [24] M. Zerner. Quelques propriétés spectrales des opérateurs positifs. J. Funct. Anal.72 (1987) 381–417. Zbl0642.47031MR886819

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.