Geodesic
Oscillating Random Walks with Two Levels of Switching
Matematičeskie trudy, Tome 6 (2003) no. 1, pp. 34-74

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We consider a Markov random walk $X_n$, $n\ge 0$, generated by the sums of independent random variables. Each successive jump of the random walk is distributed in accord with one of three laws in dependence on the location of a walking particle: within some interval $[a,b]$, to the left of the point $a$, or to the right of the point $b$. Using factorization methods, we obtain some representations for the double Laplace–Stieltjes transforms (in time and spatial variables) of the distribution of $X_n$ and find the transforms of the stationary distribution of a chain. 
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     author = {D. K. Kim and V. I. Lotov},
     title = {Oscillating {Random} {Walks} with {Two} {Levels} of {Switching}},
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D. K. Kim; V. I. Lotov. Oscillating Random Walks with Two Levels of Switching. Matematičeskie trudy, Tome 6 (2003) no. 1, pp. 34-74. http://geodesic.mathdoc.fr/item/MT_2003_6_1_a2/