Invariance principles for random walks conditioned to stay positive

Let {Sn} be a random walk in the domain of attraction of a stable law y, i.e. there exists a sequence of positive real numbers (a n) such that Sn/an converges 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. © Association des Publications de l'Institut Henri Poincaré, 2008.