Geodesic
On the Asymptotic Behavior of the Distributions of First-Passage Times, I
Matematičeskie zametki, Tome 75 (2004) no. 1, pp. 24-39

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In this paper, the asymptotic behavior of and estimates for the distribution of first-passage times for a random walk are obtained in the cases of fixed and increasing levels. In the first part of the paper, the case of zero level is studied. 
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A. A. Borovkov. On the Asymptotic Behavior of the Distributions of First-Passage Times, I. Matematičeskie zametki, Tome 75 (2004) no. 1, pp. 24-39. http://geodesic.mathdoc.fr/item/MZM_2004_75_1_a3/