This paper presents some results that are useful in the study of asymptotic approximations of boundary crossing probabilities for random walks. The main result is a refinement of an asymptotic expansion of Siegmund concerning moments of the first ladder height of random walks having small positive drift. An analysis of the covariance between the first passage time and the overshoot of a random walk over a horizontal boundary contributes to the development of the main result and is of independent interest as well. An application of these results to a "moderate deviations" approximation for the probability distribution of the time to false alarm in the cusum procedure is briefly described.
"On Moments of the First Ladder Height of Random Walks with Small Drift." Ann. Appl. Probab. 2 (3) 714 - 738, August, 1992. https://doi.org/10.1214/aoap/1177005656