Published online by Cambridge University Press: 25 March 2003
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.
To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.
To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.