Invariance principles for random walks conditioned to stay positive

Abstract

Let $\{S_n\}$ 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 $(a_n)$ such that $S_n/a_n$ converges in law to $\mathcal{Y}$. Our main result is that the rescaled process $(S_{\lfloor nt\rfloor }/a_n,t\ge 0)$, 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.

Published in at http://dx.doi.org/10.1214/07-AIHP119 the Annales de l'Institut Henri Poincaré - Probabilités et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Localization and delocalization of random interfaces
• A Local Limit Theorem for random walks conditioned to stay positive

Cited by in corpus (1)

• Invariance principles for local times at the supremum of random walks and Lévy processes