Invariance principles for random walks conditioned to stay positive

Caravenna, Francesco ; Chaumont, Loïc

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 170-190 / Harvested from Numdam

Résumé

Soit $\left\lbrace S_n \right\rbrace $ une marche aléatoire dont la loi est dans le domaine d’attraction d’une loi stable $\mathcal {Y}$, i.e. il existe une suite de réels positifs $(a_n)$ telle que $S_n/a_n$ converge en loi vers $\mathcal {Y}$. Nous montrons que le processus renormalisé $(S_{\left\lfloor nt \right\rfloor }/a_n, t\ge 0)$, une fois conditionné à rester positif, converge en loi (au sens fonctionnel) vers le processus de Lévy stable de loi $\mathcal {Y}$ conditionné à rester positif. Sous certaines hypothèses supplémentaires, nous montrons un principe d’invariance pour cette marche aléatoire tuée lorsqu’elle quitte la demi-droite positive et conditionnée à mourir en 0.

Abstract

Let $\left\lbrace S_n \right\rbrace $ be a random walk in the domain of attraction of a stable law $\mathcal {Y}$, i.e. there exists a sequence of positive real numbers $(a_n)$ such that $S_n/a_n$ converges in law to $\mathcal {Y}$. Our main result is that the rescaled process $(S_{\left\lfloor nt \right\rfloor }/a_n, t\ge 0)$, when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.

Détail

Publié le : 2008-01-01

MR 2451576 | Zbl 1175.60029 | 2 citations dans Numdam

DOI : https://doi.org/10.1214/07-AIHP119
Classification:  60G18,  60G51,  60B10

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