RANDOM WALK CONDITIONED TO STAY POSITIVE

Abstract

A random walk that is certain to visit $(0, \infty)$ has associated with it, via a suitable $h$ -transform, a Markov chain called ‘random walk conditioned to stay positive’, which is defined properly below. In continuous time, if the random walk is replaced by Brownian motion then the analogous associated process is Bessel-3. Let $\phi(x) = \log\log x$ . The main result obtained in this paper, which is stated formally in Theorem 1, is that, when the random walk has zero mean and finite variance, the total time for which the random walk conditioned to stay positive is below $x$ ultimately lies between $Lx^2/\phi(x)$ and $Ux^2\phi(x)$ , for suitable (non-random) positive $L$ and finite $U$ , as $x$ goes to infinity. For Bessel-3, the best $L$ and $U$ are identified.