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Journal of the London Mathematical Society 2003 67(1):259-272; doi:10.1112/S0024610702003708
© 2003 by London Mathematical Society
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© The London Mathematical Society

Random Walk Conditioned to Stay Positive

J. D. Biggins

Department of Probability and Statistics, Hicks Building, University of Sheffield Sheffield S3 7RH, j.biggins{at}sheffield.ac.uk

Received 28 September 2001. Revision received 20 March 2002.

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 {varphi}(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 Lx2/{varphi}(x) and Ux2{varphi}(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.


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