Geodesic
Asymptotic Analysis of Oscillating Random Walks with Two Levels of Switching
Matematičeskie trudy, Tome 8 (2005) no. 2, pp. 137-167

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We obtain complete asymptotic expansions for the prestationary distribution of an oscillating random walk $X_n$ with two levels of switching $a0$ and $b>0$ in the case when the numbers $|a|$ and $b$ increase in proportion with $\sqrt n$ as $n\to\infty$. 
@article{MT_2005_8_2_a4,
     author = {D. K. Kim},
     title = {Asymptotic {Analysis} of {Oscillating} {Random} {Walks} with {Two} {Levels} of {Switching}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {137--167},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2005_8_2_a4/}
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D. K. Kim. Asymptotic Analysis of Oscillating Random Walks with Two Levels of Switching. Matematičeskie trudy, Tome 8 (2005) no. 2, pp. 137-167. http://geodesic.mathdoc.fr/item/MT_2005_8_2_a4/