For a discrete time Markov chain, its limiting distribution is defined to be the same for all the initial distributions. A distribution over the state space is called a reversible distribution, if it satisfies the detailed balance equation.
I think a reversible distribution may not be the limiting distribution, because it may happen that a reversible distribution exists while the limiting distribution doesn't. But when the limiting distribution exist, all stationary distributions (including reversible distributions) must be the same as the limiting distribution. Please correct me if I was wrong.
Conversely, must the limiting distribution (if exists) be a reversible distribution? A counterexample will suffice. I can imagine that counterexample Markov chain will need to have the limiting distribution, and doesn't have a reversible distribution.
Thanks and regards!