A functional limit theorem for random walk conditioned to stay non-negative

In this paper we consider an aperiodic integer-valued random walk S and a process S* that is a harmonic transform of S killed when it first enters the negative half; informally, S* 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*, 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*, and a bivariate renewal theorem for the ladder time and height process, which may be of independent interest.