On the distribution of the statistic of the most powerful test in compound Markov chains

The paper deals with a sequence of outcomes realized after $n$ tests, connected into a compound Markov chain of order $s-1$ with $k$ states and with the transition probabilities of the form $1/k+\alpha/k\varphi(k)$ where $\varphi(k)$ as $k\to\infty$ and $|\alpha|<C$.
The statistic of the most powerful test for two simple hypotheses concerning the values of $\alpha$ is considered and some asymptotic properties of this statistic are studied as $n$, $k\to\infty$.