I'm having some difficulty understanding a proof in James('Chuck') Norris book on markov chains.
Let $P$ be irreducible and aperiodic, with an invariant distribution $\pi$. Let $\left(X_n\right)_{n\geq 0}$ be a discrete time Markov$\left(\lambda,P\right)$, where $\lambda$ is the initial distribution. In this setting, Chuck wants to prove that the $X_n$ converges to an equilibrium distribution.
The first part of the proof is the picture below.
1) Theorems 1.7.7 and 1.5.7 were proven for 'univariate' markov chains($X_n$ or $Y_n$), not multivariate like $W_n$. How can we apply these theorems to $W_n$? Also, what's the meaning of an invariant distribution when $\lambda$ is of dimension $I \times I$? Do I apply the usual definition to rows and columns?
Any help would be appreciated.
$W_n$ can be easily regarded as a regular (single-dimension) Markov chain. For example, say $X_n,Y_n$ share a common state space $S=\{0,1\}$. Then $W_n$ has state space $S_W=\{(0,0),\;(0,1),\;(1,0),\;(1,1)\}$. You can choose to relabel these $4$ states if you wish. So $P_W$ is a $4\times 4$ matrix and can be treated like any other Markov chain. Hence the previous theorems are valid for it.