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 (S⌊nt⌋/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.
Publié le : 2008-02-15
Classification:
Random walk,
Stable law,
Lévy process,
Conditioning to stay positive,
Invariance principle,
60G18,
60G51,
60B10
@article{1203969873,
author = {Caravenna, Francesco and Chaumont, Lo\"\i c},
title = {Invariance principles for random walks conditioned to stay positive},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {44},
number = {2},
year = {2008},
pages = { 170-190},
language = {en},
url = {http://dml.mathdoc.fr/item/1203969873}
}
Caravenna, Francesco; Chaumont, Loïc. Invariance principles for random walks conditioned to stay positive. Ann. Inst. H. Poincaré Probab. Statist., Tome 44 (2008) no. 2, pp. 170-190. http://gdmltest.u-ga.fr/item/1203969873/