Geodesic
Limit theorems and testing hypotheses on Markov chains
Diskretnaya Matematika, Tome 15 (2003) no. 4, pp. 35-65.

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We consider the optimal tests based on the likelihood ratio for discriminating between two Markov chains having a common finite phase space $\mathcal S$. Their risks are expressed in terms of probabilities of large deviations for sum of random variables defined on another Markov chain with the phase space $\mathcal S\times\mathcal S$. Both simple and composite alternatives are considered. The established asymptotic formulas for the considered risks are precise. 
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A. V. Nagaev. Limit theorems and testing hypotheses on Markov chains. Diskretnaya Matematika, Tome 15 (2003) no. 4, pp. 35-65. http://geodesic.mathdoc.fr/item/DM_2003_15_4_a2/

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