A discrete time, finite Markov chain with fixed initial state and stationary transition behavior is considered. Using Whittle's formula a large deviation result (similar to Hoeffding's result for one multinomial distribution) is obtained for the transition count matrix of a path of the chain of arbitrary length. This result is then used in the asymptotic comparison of a given sequence of tests about the transition probability matrix with a suitably constructed sequence of likelihood ratio tests. It is assumed that the sizes of these tests decrease to zero at a certain rate as the length of the observed path increases. The comparison is carried out at fixed alternatives in terms of the behavior of the ratio of type-II-error probabilities.
Luis B. Boza. "Asymptotically Optimal Tests for Finite Markov Chains." Ann. Math. Statist. 42 (6) 1992 - 2007, December, 1971. https://doi.org/10.1214/aoms/1177693067