Soit une marche aléatoire dont la loi est dans le domaine d’attraction d’une loi stable , i.e. il existe une suite de réels positifs telle que converge en loi vers . Nous montrons que le processus renormalisé , une fois conditionné à rester positif, converge en loi (au sens fonctionnel) vers le processus de Lévy stable de loi 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 be a random walk in the domain of attraction of a stable law , i.e. there exists a sequence of positive real numbers such that converges in law to . Our main result is that the rescaled process , 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.
Mots clés : random walk, stable law, Lévy process, conditioning to stay positive, invariance principle
@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}, pages = {170--190}, publisher = {Gauthier-Villars}, volume = {44}, number = {1}, year = {2008}, doi = {10.1214/07-AIHP119}, mrnumber = {2451576}, zbl = {1175.60029}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP119/} }
TY - JOUR AU - Caravenna, Francesco AU - Chaumont, Loïc TI - Invariance principles for random walks conditioned to stay positive JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2008 SP - 170 EP - 190 VL - 44 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP119/ DO - 10.1214/07-AIHP119 LA - en ID - AIHPB_2008__44_1_170_0 ER -
%0 Journal Article %A Caravenna, Francesco %A Chaumont, Loïc %T Invariance principles for random walks conditioned to stay positive %J Annales de l'I.H.P. Probabilités et statistiques %D 2008 %P 170-190 %V 44 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP119/ %R 10.1214/07-AIHP119 %G en %F 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/articles/10.1214/07-AIHP119/
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