Invariance principles for random walks conditioned to stay positive

Francesco Caravenna; Loïc Chaumont

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 1, page 170-190
  • ISSN: 0246-0203

Abstract

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Let {Snbe a random walk in the domain of attraction of a stable law 𝒴 , i.e. there exists a sequence of positive real numbers ( an) such that Sn/anconverges in law to 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.

How to cite

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Caravenna, Francesco, and Chaumont, Loïc. "Invariance principles for random walks conditioned to stay positive." Annales de l'I.H.P. Probabilités et statistiques 44.1 (2008): 170-190. <http://eudml.org/doc/77960>.

@article{Caravenna2008,
abstract = {Let \{Snbe 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/anconverges 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.},
author = {Caravenna, Francesco, Chaumont, Loïc},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk; stable law; Lévy process; conditioning to stay positive; invariance principle},
language = {eng},
number = {1},
pages = {170-190},
publisher = {Gauthier-Villars},
title = {Invariance principles for random walks conditioned to stay positive},
url = {http://eudml.org/doc/77960},
volume = {44},
year = {2008},
}

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
PB - Gauthier-Villars
VL - 44
IS - 1
SP - 170
EP - 190
AB - Let {Snbe 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/anconverges 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.
LA - eng
KW - random walk; stable law; Lévy process; conditioning to stay positive; invariance principle
UR - http://eudml.org/doc/77960
ER -

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