Geodesic
Random walks on a~halfaxis. I.~Boundary problems and ergodic theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 1, pp. 45-58

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We investigate the Markov right continuous homogeneous Feller processes with the state space $\{1,\dots,d\}\times[0,\infty)$. It is assumed that up to the moment of the first entrance in the set of states $\{(i,0)\colon i=1,\dots,d\}$ the process develops like homogeneous Markov process which is second-component homogeneous without negative jumps of the second component. In § 1 the existence theorem is proved and all such processes are described. In § 2 some functionals associated with the moment when the second component leaves an interval are studied. The results of § 2 are then used in the proof of the ergodic theorem. 
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     author = {V. M. \v{S}urenkov},
     title = {Random walks on a~halfaxis. {I.~Boundary} problems and ergodic theorem},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {45--58},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a3/}
}
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V. M. Šurenkov. Random walks on a~halfaxis. I.~Boundary problems and ergodic theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 1, pp. 45-58. http://geodesic.mathdoc.fr/item/TVP_1981_26_1_a3/