The scaling limits of a heavy tailed Markov renewal process

Sohier, Julien

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

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.

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.

Classification: 60F77, 60K15, 60K20, 60K05, 82B27
Keywords: Heavy tailed Markov renewals processes, scaling limits, fluctuation theory for random walks, regenerative sets

[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