Geodesic
Ergodicity properties of conditional Markov chains
Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 3, pp. 547-557 
Z. I. Bezhaeva. Ergodicity properties of conditional Markov chains. Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 3, pp. 547-557. http://geodesic.mathdoc.fr/item/TVP_1974_19_3_a6/
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     author = {Z. I. Bezhaeva},
     title = {Ergodicity properties of conditional {Markov} chains},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {547--557},
     year = {1974},
     volume = {19},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1974_19_3_a6/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\zeta_t(\omega)=(\xi_t(\omega),\eta_t(\omega))$, $t=1,2,\dots,$ be a finite homogeneous Markov chain. If $\eta_1(\omega),\dots,\eta_n(\omega)$ are fixed, $\xi_t(\omega)$, $t=1,\dots,n,$ is a so called conditional Markov chain. In this article, properties of trajectories of the conditional Markov chain and ergodicity properties of it are investigated.