Geodesic
Weak convergence of matrices of transition probabilities for the conditioned Markov chains
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 1, pp. 57-66

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Let $\zeta_t=(\xi_t,\eta_t)$ be a two-dimensional countable Markov chain. The component $\xi_t\,(t=1\div n)$ may be considered as a conditioned Markov chain with respect to the conditional probability measure $\mathbf P\{\cdot\mid\eta_1,\dots,\eta_n\}$. We prove that under some assumptions all components of the matrix of transition probabilities of conditioned Markov chain converge weakly to the corresponding limits when $n\to\infty$. 
@article{TVP_1982_27_1_a5,
     author = {Z. I. Be\v{z}aeva},
     title = {Weak convergence of matrices of transition probabilities for the conditioned {Markov} chains},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {57--66},
     publisher = {mathdoc},
     volume = {27},
     number = {1},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_1_a5/}
}
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Z. I. Bežaeva. Weak convergence of matrices of transition probabilities for the conditioned Markov chains. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 1, pp. 57-66. http://geodesic.mathdoc.fr/item/TVP_1982_27_1_a5/