A Functional Limit Theorem for Random Walk Conditioned to Stay Non-Negative

In this paper we consider an aperiodic integer-valued randomwalk S and a process S* that is a harmonic transform of S killedwhen it first enters the negative half; informally, S* is ‘Sconditioned to stay non-negative’. If S is in the domainof attraction of the standard normal law, without centring,a suitably normed and linearly interpolated version of S convergesweakly to standard Brownian motion, and our main result is thatunder the same assumptions a corresponding statement holds forS*, the limit of course being the three-dimensional Bessel process.As this process can be thought of as Brownian motion conditionedto stay non-negative, in essence our result shows that the interchangeof the two limit operations is valid. We also establish somerelated results, including a local limit theorem for S*, anda bivariate renewal theorem for the ladder time and height process,which may be of independent interest.