A FUNCTIONAL LIMIT THEOREM FOR RANDOM WALK CONDITIONED TO STAY NON-NEGATIVE

Abstract

In this paper we consider an aperiodic integer-valued random walk $S$ and a process $S^{\ast }$ that is a harmonic transform of $S$ killed when it first enters the negative half; informally, $S^{\ast }$ is ‘$S$ conditioned to stay non-negative’. If $S$ is in the domain of attraction of the standard normal law, without centring, a suitably normed and linearly interpolated version of $S$ converges weakly to standard Brownian motion, and our main result is that under the same assumptions a corresponding statement holds for $S^{\ast }$, the limit of course being the three-dimensional Bessel process. As this process can be thought of as Brownian motion conditioned to stay non-negative, in essence our result shows that the interchange of the two limit operations is valid. We also establish some related results, including a local limit theorem for $S^{\ast }$, and a bivariate renewal theorem for the ladder time and height process, which may be of independent interest.