Time Periodic Homogeneous Markov Chain I want to find a textbook or survey article reference with a treatment of discrete-time, inhomogeneous, yet time periodic, markov chains on finite state spaces.
Elaboration: I have an inhomogeneous chain with transition matrices $P^{(t)}$, where $P^{(t + n)} = P^{(t)}$. Let $S$ be the state space. What would like to do is capture this behavior on a larger state space with a time-homogeneous chain. To do this, consider the bigger state space $\hat{S} = S \times [1,n]$, where $[1,n] = \{1, 2, \cdots, n\}$ is the 'time coordinate'. The transition matrix $\hat{P}$ on this chain acts like
$$
(i,t) \rightarrow (j, t+1)
$$
with probability $(P^{(t)})_{ij}$. Writing the full transition matrix in $S$-blocks, this should read
$$
\hat{P} = \left(\begin{array}{c c c c c} 
0 & P^{(1)} & 0 & \cdots & 0  \\
\vdots & \ddots & P^{(2)} & \ddots & \vdots \\
0 & \cdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & 0 &  P^{(n-1)} \\
P^{(n)} & 0 & \cdots & 0 & 0 
\end{array} \right)
$$

Operations Research: An Introduction by Taha covers this I believe.
