Geodesic
Conditional Markov chains with denumerable set of states
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 556-565

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Let $\zeta_t=(\xi_t,\eta_t)$ ($t=1,2,\dots$) be a denumerable homogeneous Markov chain. If $\eta_1,\dots,\eta_n$ are fixed, $\xi_t$ ($t=1,2,\dots,n$) is a conditional Markov chain. In this paper, ergodicity properties of such chains are proved. 
@article{TVP_1977_22_3_a8,
     author = {Z. I. Be\v{z}aeva},
     title = {Conditional {Markov} chains with denumerable set of states},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {556--565},
     publisher = {mathdoc},
     volume = {22},
     number = {3},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_3_a8/}
}
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Z. I. Bežaeva. Conditional Markov chains with denumerable set of states. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 556-565. http://geodesic.mathdoc.fr/item/TVP_1977_22_3_a8/