Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Invariance principles for random walks conditioned to stay positive

Francesco Caravenna and Loïc Chaumont

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Abstract

Let {Sn} 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 (an) such that Sn/an converges in law to $\mathcal{Y}$. Our main result is that the rescaled process (Snt/an, t≥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.

Résumé

Soit {Sn} 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 (an) telle que Sn/an converge en loi vers $\mathcal{Y}$. Nous montrons que le processus renormalisé (Snt/an, t≥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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 44, Number 1 (2008), 170-190.

Dates
First available in Project Euclid: 25 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1203969873

Digital Object Identifier
doi:10.1214/07-AIHP119

Mathematical Reviews number (MathSciNet)
MR2451576

Zentralblatt MATH identifier
1175.60029

Subjects
Primary: 60G18: Self-similar processes 60G51: Processes with independent increments; Lévy processes 60B10: Convergence of probability measures

Keywords
Random walk Stable law Lévy process Conditioning to stay positive Invariance principle

Citation

Caravenna, Francesco; Chaumont, Loïc. Invariance principles for random walks conditioned to stay positive. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 1, 170--190. doi:10.1214/07-AIHP119. https://projecteuclid.org/euclid.aihp/1203969873


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