The scaling limits of a heavy tailed Markov renewal process

Sohier, Julien

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 483-505.

Dans cet article, nous considérons des processus de renouvellement markovien à queues lourdes. Nous montrons que, convenablement renormalisés, ils convergent vers l’ensemble régénératif d’indice $\alpha$. Nous appliquons ces résultats à un modèle d’accrochage dans une bande. Dans ce modèle, une marche aléatoire $S$, contrainte à rester au-dessus d’un mur, est récompensée ou pénalisée lorsqu’est atteinte la bande $[0,\infty )\times [0,a]$ où $a$ est un réel strictement positif. La convergence que nous établissons permet de caractériser les limites d’échelle de ce modèle au point critique.

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.

[1] S. Asmussen. Applied Probability and Queues, 2nd edition. Applications of Stochastic Modelling and Applied Probability 51. Springer, New York, 2003. | MR | Zbl

[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. | MR | Zbl

[3] A. N. Borodin and P. Salminen. Handbook of Brownian Motion - Facts and Formulae, 2nd edition. Probability and Its Applications. Birkhäuser, Basel, 2002. | MR | Zbl

[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. | MR | Zbl

[5] F. Caravenna and J.-D. Deuschel. Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction. Ann. Probab. 37 (2009) 903-945. | MR | Zbl

[6] F. Caravenna, G. Giacomin and L. Zambotti. Tightness conditions for polymer measures. Preprint, 2007. Available at arXiv.org:math/0702331.

[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). | MR | Zbl

[8] F. Caravenna, G. Giacomin and L. Zambotti. Infinite volume limits of polymer chains with periodic charges. Markov Process. Related Fields 13 (2007) 697-730. | MR | Zbl

[9] E. Çinlar. Some joint distributions for Markov renewal processes. Aust. N. Z. J. Stat. 10 (1968) 8-20. | MR | Zbl

[10] M. Cranston, L. Koralov, S. Molchanov and B. Vainberg. Continuous model for homopolymers. J. Funct. Anal. 256 (2009) 2656-2696. | MR | Zbl

[11] J.-D. Deuschel, G. Giacomin and L. Zambotti. Scaling limits of equilibrium wetting models in $(1+1)$-dimension. Probab. Theory Related Fields 132 (2005) 471-500. | MR | Zbl

[12] R. A. Doney. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 (1997) 451-465. | MR | Zbl

[13] R. A. Doney. Local behavior of first passage probabilities. Probab. Theory Related Fields 5 (2010) 299-315. | MR | Zbl

[14] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York, 1971. | MR | Zbl

[15] P. J. Fitzsimmons, B. Fristedt and B. Maisonneuve. Intersections and limits of regenerative sets. Z. Wahrsch. Verw. Gebiete 70 (1985) 157-173. | MR | Zbl

[16] G. Giacomin. Random Polymer Models. Imperial College Press, London, 2007. | MR | Zbl

[17] P. E. Greenwood and W. Wefelmeyer. Empirical estimators for semi-Markov processes. Math. Meth. Statist. 5 (1996) 299-315. | MR | Zbl

[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. | MR | Zbl

[19] R. Pyke. Markov renewal processes with finitely many states. Ann. Math. Statist. 32 (1961) 1243-1259. | MR | Zbl

[20] R. Pyke and R. Schaufele. Limit theorems for Markov renewal processes. Ann. Math. Statist. 35 (1964) 1746-1764. | MR | Zbl

[21] W. L. Smith. Regenerative stochastic processes. Proc. R. Soc. Lond. Ser. A 232 (1955) 6-31. | MR | Zbl

[22] J. Sohier. A functional limit convergence towards Brownian excursion. Preprint, 2010. Available at arXiv.org:1012.0118.

[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] M. Zerner. Quelques propriétés spectrales des opérateurs positifs. J. Funct. Anal. 72 (1987) 381-417. | MR | Zbl