Random Walk Conditioned to Stay Positive

A random walk that is certain to visit (0, ) has associatedwith it, via a suitable h-transform, a Markov chain called ‘randomwalk conditioned to stay positive’, which is defined properlybelow. In continuous time, if the random walk is replaced byBrownian motion then the analogous associated process is Bessel-3.Let (x) = log log x. The main result obtained in this paper,which is stated formally in Theorem 1, is that, when the randomwalk has zero mean and finite variance, the total time for whichthe random walk conditioned to stay positive is below x ultimatelylies between Lx²/(x) and Ux²(x), for suitable (non-random) positiveL and finite U, as x goes to infinity. For Bessel-3, the bestL and U are identified.