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 (21)

• An invariance principle to Ferrari-Spohn diffusions
• On Wiener-Hopf factors for stable processes
• A functional limit convergence towards brownian excursion
• At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift
• A revisited proof of the Seneta-Heyde norming for branching random walks under optimal assumptions
• On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk
• Random walks with unbounded jumps among random conductances II: Conditional quenched CLT
• Range and critical generations of a random walk on Galton-Watson trees
• Extreme order statistics of random walks
• Simple peeling of planar maps with application to site percolation
• Scaling limit of the path leading to the leftmost particle in a branching random walk
• Conditional and uniform quenched CLTs for one-dimensional random walks among random conductances
• Markovian explorations of random planar maps are roundish
• A conditioned local limit theorem for non-negative random matrices
• Invariance principles for local times at the supremum of random walks and Lévy processes
• Seneta-Heyde norming for branching random walks with $\alpha$-stable spine
• Local equilibrium in planar non interacting particle systems
• How many families survive for a long time
• An invariance principle for random walk bridges conditioned to stay positive
• Intermediately subcritical branching process in random environment: the initial stage of the evolution
• The two-sided exit problem for a random walk on $\mathbb{Z}$ and having infinite variance II