RANDOM WALKS WITH NEGATIVE DRIFT CONDITIONED TO STAY POSITIVE

Let {Xk: k ? 1 } be a sequence of independent, identically distributed random variables with EXI = p 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 p < 0 and conditional limits S* (p), and show that S* (p), properly normalized, converges to a gamma distribution of second order as s /; 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.