Discrete Time Markov Chains

The Markov chain \((X_i)_{i \in \mathbb {N}}\) can be seen as representing the evolution of a system that moves from one state to another in a random fashion. We have assumed that \(S \subset \mathbb {R}\), but it may be convenient to consider in some situations a general finite set S. In this case \(X_i\) are not random numbers, but random entities.