Invariance principles for random walks conditioned to stay positive

Let {S-n} 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 S-n/a(n) converges in law to y. Our main result is that the rescaled process (S-[nt]/a(n), t >= 0), when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Levy 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.