Random walks with negative drift conditioned to stay positive

<jats:p>Let {<jats:italic>X<jats:sub>k</jats:sub></jats:italic>: <jats:italic>k</jats:italic> ≧ 1} be a sequence of independent, identically distributed random variables with <jats:italic>EX</jats:italic><jats:sub>1</jats:sub> = <jats:italic>μ</jats:italic> < 0. Form the random walk {<jats:italic>S<jats:sub>n</jats:sub></jats:italic>: <jats:italic>n</jats:italic> ≧ 0} by setting <jats:italic>S</jats:italic><jats:sub>0</jats:sub> = 0, <jats:italic>S<jats:sub>n</jats:sub></jats:italic> = <jats:italic>X</jats:italic><jats:sub>1</jats:sub> + … + <jats:italic>X<jats:sub>n</jats:sub>, n ≧</jats:italic> 1. Let <jats:italic>T</jats:italic> denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of <jats:italic>X</jats:italic><jats:sub>1</jats:sub>) that <jats:italic>S<jats:sub>n</jats:sub></jats:italic>, conditioned on <jats:italic>T</jats:italic> > <jats:italic>n</jats:italic> converges weakly to a limit random variable, <jats:italic>S</jats:italic>∗, and to find the Laplace transform of the distribution of <jats:italic>S∗</jats:italic>. We also investigate a collection of random walks with mean <jats:italic>μ</jats:italic> < 0 and conditional limits <jats:italic>S∗</jats:italic> (<jats:italic>μ</jats:italic>), and show that <jats:italic>S</jats:italic>∗ (<jats:italic>μ</jats:italic>), properly normalized, converges to a gamma distribution of second order as <jats:italic>μ</jats:italic> ↗ 0. These results have applications to the <jats:italic>GI</jats:italic>/<jats:italic>G</jats:italic>/1 queue, collective risk theory, and the gambler's ruin problem.</jats:p>