Someone told me that this discrete stochastic process $(n,x_n)$ is always a time homogeneous Markov chain (where $(x_n)$ is a discrete Markov chain).
What do you think of that proposition ? It seems false to me, because I don't see how adding $n$ to the process makes it T.H.
Otherwise, do you know how to transform any Markov chain into a time homogeneous one ?
Every state in the new Markov chain has the time included in the state itself. Therefore, the transition probabilities only depend on the states.
That's what makes it time-homogeneous, but in a rather brutish way. For instance, it will turn a finite-state markov chain into an infinite-state markov chain, and every state in the new chain is transient because they can be visited at most once.