Invariance principles for random walks conditioned to stay positive

Francesco Caravenna (1) , Loïc Chaumont (2)

1 Institut für Mathematik [Zürich]
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Let $\{S_n\}$ be a random walk in the domain of attraction of a stable law $\cY$, i.e. there exists a sequence of positive real numbers $(a_n)$ such that $S_n/a_n$ converges in law to $\cY$. Our main result is that the rescaled process $(S_{\lfloor nt \rfloor}/a_n,\,t\ge0)$, when conditioned to stay positive for all the time, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive in the same sense. 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: invariance principle; Random walk; stable law; Lévy process; conditioning to stay positive; invariance principle.

Domaines: Probabilités [math.PR]

Francesco Caravenna, Loïc Chaumont. Invariance principles for random walks conditioned to stay positive. 2006. ⟨hal-00019049⟩