The steps of a one-dimensional random walk are positive and occur randomly in time at a fixed mean rate. The sizes of the steps are independent and the size of each step has the same given probability distribution. The distribution of the time to reach a fixed barrier is obtained and approximations to its moments are derived. The results are extended to the case in which the barrier and the random walk process converge at a constant rate.
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Mercer et al. (1959) studied this question.