A sufficient condition for a unique invariant distribution of a higher-order Markov chain

We derive a sufficient condition for a kth order homogeneous Markov chain Z with finite alphabet Z to have a unique invariant distribution on Zk. Specifically, let X be a first-order, stationary Markov chain with finite alphabet X and a single recurrent class, let g:X→Z be non-injective, and define the (possibly non-Markovian) process Y:=g(X) (where g is applied coordinate-wise). If Z is the kth order Markov approximation of Y, its invariant distribution is unique. We generalize this to non-Markovian processes X.