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 $\cY$, 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.
Loading...