A Sufficient Condition for a Unique Invariant Distribution of a Higher-Order Markov Chain

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