If a discrete markov chain is stationary (as far as I know: doesn't modify itself with time), irreducible (doesn't have transient states) and aperiodic (no periodic states), is it positive recurrent?
This answer might be answered or not, the problem is: I don't know whether it can.
A chain is positive recurrent if mean recurrence time is finite, but it seems to me that I don't know how many states are there in a generic discrete markov chain (could it be infinite states?).