Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 28K / b

A continuous-time Markov chain is irreducible if every state can reach every other state with positive probability. A state $i$ is recurrent if, after leaving $i$, the chain returns to $i$ with probability one; an irreducible chain is recurrent if one, and hence every, state is recurrent. A recurrent state is positive recurrent if its expected continuous first return time is finite, and an irreducible chain is positive recurrent when all its states are.