# Random walk conditioned to stay positive

by J.D.Biggins

2000 Mathematics Subject Classification: Primary 60J10; secondary 60G50

## Abstract

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.

## Availability: pdf file or
ps file of present version;
revised 8 March 2002

Submitted to LMS

Other publications by J.D.
Biggins

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