2000 Mathematics Subject Classification: Primary 60J10; secondary 60G50
Given a random walk with mean zero and finite variance, the progress to infinity of the associated random walk conditioned to stay positive is studied through the sample path representation of Tanaka (1989). Specifically, if $D(x)$ is the time that the process spends below $x$ and $\phi(x)= \log \log x$ then, as $x$ goes to infinity, $D(x) / x^2 $ ultimately lies between $L / \phi(x)$ and $U \phi (x)$ for suitable (non-random) positive $L$ and finite $U$. The Bessel-3 is one continuous analogue; for it the best $L$ and $U$ are identified.
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