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Let \(\; X_1, X_2, \ldots \;\) be a homogeneous Markov chain with values on a finite state space \(\; D \;\) and transition probability matrix \(\; P = (p_{ij}), \: i, j \in D \;\). \(\; P \;\) determines whether the sequence can go from any state \(\; i \;\) to any other state \(\; j \;\) in a single step, and more generally \(\; P^{n} \;\) tells us whether a path exists from a state \(\; i \;\) to another state \(\; j \;\) in \(\; n \;\) steps. We denote the existence of a path between \(\; i \;\) and \(\; j \;\) as \(\; i \rightsquigarrow j \;\), and the lack of a path as \(\; i \not\rightsquigarrow j \;\)

A state \(\; i \;\) is called transient if it is eventually abandoned, never to return to it again. This is because there exists at least one state \(\; j \;\) for which there are outward paths, \(\; i \rightsquigarrow j \;\), but there are no returning paths, \(\; j \not\rightsquigarrow i \;\), so the chain will eventually move away from \(\; i \;\) forever.

A state \(\; i \;\) is called recurrent if it is forever visited over and over again, infinite times. This is because for every outward path \(\; i \rightsquigarrow j \;\) there exists a returning path \(\; j \not\rightsquigarrow i \;\), so the chain will always eventually come back sooner or later.

If a recurrent state \(\; i \;\) has a path towards another state \(\; j \;\), then \(\; j \;\) must also be recurrent (or else there would be trajectories that would take the Markov chain away from both states forever, making both of them transient).

Similarly, if a state \(\; i \;\) has a path towards a transient state \(\; j \;\), then \(\; i \;\) must also be transient (since the second state has trajectories that take the Markov chain away forever).
Therefore, the state space \(\; D \;\) can be partitioned into distinct regions of interconnected states that are all either recurrent or all transient. Each of the recurrent regions is called a recurrence class.