Can a finite state Markov chain have essential transient state?
I have found out an example for an infinite state one and I have the intuition (I may be wrong) that for a finite state space .. This isn't possible... But I am not being able to prove it..
For a finite state Markov Chain, suppose $i$ is an essential transient state. Then clearly as $i$ is not absorbing, it must lead to some state $j\neq i$.
Since $i$ is essential, and $i\to j$ we have $j\to i$. So we can look at the equivalence class of $i$.
Of course $j$ cannot be recurrent, because then it would mean $i$ is recurrent. Thus $j$ is transient. So our equivalence class in a finite state MC has only transient states. This is a contradiction.