$X_n, n> 0$ is a Markov Chain, how to interpret $Z_n = (X_n,X_{n+1}), n > 0$?

Question

Am a newbie to Markov Chain. So, this might be incredibly naive/stupid question. If $X_n, \, n > 0$ is MC, am having difficulty imagining/interpreting process $Z_n = (X_n,X_{n+1}), n > 0$. I have to show if $Z_n$ is MC and if it is, find the transition matrix and stationary distribution ($P$ is transition matrix of $X_n$ and $\pi$ is its stationary distribution).

Thanks

Answer 1

$$Q((x,y),(x',y'))=\mathbf 1_{y=x'}\cdot q(y,y')$$ $$ \Pi(x,y)=\pi(x)\cdot q(x,y)$$