You should also check the classic references to see whether your 2 matrices are
consistent with your null of stationarity:
Anderson, T. W. and L. A. Goodman. 1957. Statistical inference about Markov
chains. Annals of Mathematical Statistics 28: 89-110.
Billingsley, P. 1961. Statistical methods in Markov chains. Ann. Math.
Statist. 32: 12-40.
Chatfield, C. 1973. Statistical inference regarding Markov chain models.
Appl. Statist. 22: 7-20.
Kullback, S., M. Kupperman, and H. H. Ku. 1962. Tests for contingency tables
and Markov chains. Technometrics 4: 573-608.
Sandland, R. L. 1976. Applications of methods of testing for independence
between two Markov chains. Biometrics 32: 629-636.
Usher, M. B. 1979. Markovian approaches to ecological succession. Journal
of Animal Ecology 48: 413-426. [Applies some of the likelihood models cited
above to test stationarity in a Markov process with some simulated and real
data]
.
.
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