Invariance principles for random walks conditioned to stay positive
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, p. 170-190
Soit S n une marche aléatoire dont la loi est dans le domaine d’attraction d’une loi stable 𝒴, i.e. il existe une suite de réels positifs (a n ) telle que S n /a n converge en loi vers 𝒴. Nous montrons que le processus renormalisé (S nt /a n ,t0), une fois conditionné à rester positif, converge en loi (au sens fonctionnel) vers le processus de Lévy stable de loi 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.
Let S n be a random walk in the domain of attraction of a stable law 𝒴, i.e. there exists a sequence of positive real numbers (a n ) such that S n /a n converges in law to 𝒴. Our main result is that the rescaled process (S nt /a n ,t0), 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.
@article{AIHPB_2008__44_1_170_0,
     author = {Caravenna, Francesco and Chaumont, Lo\"\i c},
     title = {Invariance principles for random walks conditioned to stay positive},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {1},
     year = {2008},
     pages = {170-190},
     doi = {10.1214/07-AIHP119},
     zbl = {1175.60029},
     mrnumber = {2451576},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2008__44_1_170_0}
}
Caravenna, Francesco; Chaumont, Loïc. Invariance principles for random walks conditioned to stay positive. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 170-190. doi : 10.1214/07-AIHP119. http://www.numdam.org/item/AIHPB_2008__44_1_170_0/

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