Random walks with negative drift conditioned to stay positive

Let {Xk : k ≧ 1} be a sequence of independent, identically distributed random variables with EX 1 = μ < 0. Form the random walk {Sn : n ≧ 0} by setting S 0 = 0, Sn = X 1 + … + Xn, n ≧ 1. Let T 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 X 1) that Sn , conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.