Geodesic
Limit theorems for conditional Markov chains
Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 3, pp. 437-445 
Z. I. Bežaeva. Limit theorems for conditional Markov chains. Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 3, pp. 437-445. http://geodesic.mathdoc.fr/item/TVP_1971_16_3_a1/
@article{TVP_1971_16_3_a1,
     author = {Z. I. Be\v{z}aeva},
     title = {Limit theorems for conditional {Markov} chains},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {437--445},
     year = {1971},
     volume = {16},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1971_16_3_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\zeta_k(\omega)=(\xi_k(\omega),\eta_k(\omega))$ ($k=1,2,\dots$) be a finite homogeneous Markov chain. If $\eta_1(\omega),\dots,\eta_n(\omega)$ are fixed, $\xi_k$ ($k=1,\dots,n$) are known to be a so-called conditional Markov chain. In this paper, the law of large numbers and the central limit theorem for the conditional Markov chain are obtained.