Geodesic
Asymptotic analysis of distributions in the problems with two boundaries.~II
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 4, pp. 873-879

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Let the conditions of the first part of the paper are satisfied. We obtain the complete asymptotic expansions of the probabilities \begin{gather*} \mathbf P\{S_n=k,\,N>n\},\qquad k\in(-a,b),\\ \mathbf P\{S_N=k,\,N=n\},\qquad k\notin(-a,b), \end{gather*} for the case $a=a(n)=o(n)$, $b=b(n)=o(n)$, $a\to\infty$, $b\to\infty$, $a+b\ge Cn^{1/2}$. 
@article{TVP_1979_24_4_a20,
     author = {V. I. Lotov},
     title = {Asymptotic analysis of distributions in the problems with two {boundaries.~II}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {873--879},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_4_a20/}
}
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V. I. Lotov. Asymptotic analysis of distributions in the problems with two boundaries.~II. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 4, pp. 873-879. http://geodesic.mathdoc.fr/item/TVP_1979_24_4_a20/