Inference about stationary distributions of Markov chains based on divergences with observed frequencies

For data generated by stationary Markov chains there are considered estimates of chain parameters minimizing ^-divergences between theoretical and empirical distributions of states. Consistency and asymptotic normality are established and the asymptotic covariance matrices are evaluated. Testing of hypotheses about the stationary distributions based on (^-divergences between the estimated and empirical distributions is considered as well. Asymptotic distributions of (^-divergence test statistics are found, enabling to specify asymptotically o>level tests.