The scaling limits of a heavy tailed Markov renewal process
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, p. 483-505

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.

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 α. 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,)×[0,a]a est un réel strictement positif. La convergence que nous établissons permet de caractériser les limites d’échelle de ce modèle au point critique.

DOI : https://doi.org/10.1214/11-AIHP456
Classification:  60F77,  60K15,  60K20,  60K05,  82B27
Keywords: Heavy tailed Markov renewals processes, scaling limits, fluctuation theory for random walks, regenerative sets
@article{AIHPB_2013__49_2_483_0,
     author = {Sohier, Julien},
     title = {The scaling limits of a heavy tailed Markov renewal process},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {2},
     year = {2013},
     pages = {483-505},
     doi = {10.1214/11-AIHP456},
     zbl = {1271.60095},
     mrnumber = {3088378},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_2_483_0}
}
Sohier, Julien. The scaling limits of a heavy tailed Markov renewal process. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 483-505. doi : 10.1214/11-AIHP456. http://www.numdam.org/item/AIHPB_2013__49_2_483_0/

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