Recurrence and transience of states

Summary

A state of a Markov chain is recurrent if the chain, started there, returns to it with probability one, and transient otherwise. A recurrent state is visited infinitely often, while a transient state is visited only finitely often, almost surely. Recurrent states are further split into positive recurrent and null recurrent, according to whether the mean return time is finite. In an irreducible chain all states have the same type, which decides whether a stationary distribution exists.