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.

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

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