In a Markov chain (you can add additional conditions here, such as discrete-time, homogeneous, finite-state, .... But the less additional condition, the better ), what sufficient and/or necessary condition can make every initial distribution have a limit distribution?
Note that here the limit distributions for different initial distributions may be different. Added: What I was thinking when posting the question is to include the case when there does not exist the limiting distribution same for all initial distributions, but there exists a limit distribution for every initial distribution.
Thanks and regards!
My question comes from my comment to Michael Hardy's reply.