Fisher information determinant and stochastic complexity for Markov models

We study Fisher information of stationary Markov models with a finite alphabet. In particular, we derive the Fisher information determinant of expectation parameter eta, which is defined as expectation of Markov type. The Fisher information determinant with respect to Markov kernel parameter (conditional probabilities) is easy to find, while it is not so with respect to the expectation parameter eta nor the natural parameter thetas. Note that thetas and eta are of special importance for exponential families including Markov models.