Looking for litterature on Markov chains

I am looking for a test for reducing the order of a markov chain with state space {0,1} from k to 0. The chain is observed only once for length T.

I have looked at Hoel (1954) "A Test for Markoff Chains A TEST FOR MARKOFF CHAINS". But it only covers sequential reduction, from K to K-1.

Any one got a source i could look too ?

Thank you!