Limit theorems and testing hypotheses on Markov chains

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.