Convergence of a Second Order Markov Chain

In this paper, we consider convergence properties of a second order Markov chain. Similar to a column stochastic matrix is associated to a Markov chain, a so called {\em transition probability tensor} $P$ of order 3 and dimension $n$ is associated to a second order Markov chain with $n$ states. For this $P$, define $F_P$ as $F_P(x):=Px^{2}$ on the $n-1$ dimensional standard simplex $\Delta_n$. If 1 is not an eigenvalue of $\nabla F_P$ on $\Delta_n$ and $P$ is irreducible, then there exists a unique fixed point of $F_P$ on $\Delta_n$. In particular, if every entry of $P$ is greater than $\frac{1}{2n}$, then 1 is not an eigenvalue of $\nabla F_P$ on $\Delta_n$. Under the latter condition, we further show that the second order power method for finding the unique fixed point of $F_P$ on $\Delta_n$ is globally linearly convergent and the corresponding second order Markov process is globally $R$-linearly convergent.