Necessary and Sufficient Conditions for Stability of Finite State Markov Chains

This note considers finite state Markov chains which overlap supports. While the overlapping supports condition is known to be necessary and sufficient for stability of these chains, the result is typically presented in a more general context. As such, one objective of the note is to provide an exposition, along with simple proofs corresponding to the finite case. Second, the note provides an additional equivalent condition which should be useful in applications.