When does the stationary distribution exist for a Markov chain?

Question

I am trying to find (ideally) an if and only if statement stating the existence of a stationary distribution for a time-discrete, finite state Markov chain.

So far I have only found sufficient conditions involving irreducibility of a Markov chain but not necessary and sufficient.

Answer 1

There is always a stationary distribution for any finite state (time-homogeneous) Markov chain. We normally assume irreducibility to ensure uniqueness, not existence. See Finite State Markov Chain Stationary Distribution.