In a discrete time Markov chain, consider an irreducible/communicating class,
Are the probabilities of ever transition between any two states within the class the same?
If the class is recurrent, the probabilities of each state ever transitioning back to itself are always 1 and thus the same. Here I wonder about the general case for transition between any two states or/and the class may not be recurrent.
Are the expected times of transition between any two states within the class the same?
If the class is transient or null positive, the expected times of each state transitioning back to itself are always $\infty$ and thus the same. Here I wonder about the general case for transition between any two states or/and the class may not be transient or null positive.
If the answers are no, are there some other cases where the probabilities/expected times are the same for all the transitions between any two states in an irreducible class?
Thanks and regards!