Geodesic
Stability theorems and the second-order asymptotics in threshold phenomena for boundary functionals of random walks
Matematičeskie trudy, Tome 19 (2016) no. 1, pp. 46-69.

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In Section 1, we prove stability theorems for a series of boundary functionals of random walks. In Section 2, we suggest a new simpler proof of the theorem on threshold phenomena for the distribution of the maximum of the consecutive sums of random variables. In Section 3, we find the second-order asymptotics for this distribution under the assumption that the third moments of the random variables exist. 
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A. A. Borovkov. Stability theorems and the second-order asymptotics in threshold phenomena for boundary functionals of random walks. Matematičeskie trudy, Tome 19 (2016) no. 1, pp. 46-69. http://geodesic.mathdoc.fr/item/MT_2016_19_1_a1/

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