Geodesic
Asymptotic expansions for the distribution of the sojourn time of a~random walk on a~half-axis
Informatics and Automation, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 154-164

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A complete asymptotic expansion for $n\to\infty$ is obtained in a local limit theorem for the distribution of the sojourn time of a random walk with zero drift in the set $(b,\infty)$ during $n$ steps. Here $b=b(n)\to\infty$, $b(n)=o(n)$, and Cramér-type conditions are imposed on the distribution of jumps of the walk. 
@article{TRSPY_2013_282_a12,
     author = {V. I. Lotov},
     title = {Asymptotic expansions for the distribution of the sojourn time of a~random walk on a~half-axis},
     journal = {Informatics and Automation},
     pages = {154--164},
     publisher = {mathdoc},
     volume = {282},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2013_282_a12/}
}
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V. I. Lotov. Asymptotic expansions for the distribution of the sojourn time of a~random walk on a~half-axis. Informatics and Automation, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 154-164. http://geodesic.mathdoc.fr/item/TRSPY_2013_282_a12/