Invariant Distribution of a Higher-Order Markov Chain (Pt. 2)

Some time ago I was looking into the uniqueness of the invariant distribution of a second-(or higher-)order Markov chain, and I was running into problems. After a few weeks of work, a manuscript was ready in which I found some useful sufficient conditions for this invariant distribution to be unique. The manuscript is available on arXiv, and published in the Statistics and Probability Letters. You can access the full-text of the manuscript for free until October 1st 2017 (afterwards, the arXiv version remains available, of course). I'm happy to get feedback, so please don't hold back :)!