The theorem is "If a transition matrix for an irreducible Markov chain with a ﬁnite state space S is doubly stochastic, its (unique) invariant measure is uniform over S."
If a Markov Chain has a doubly-stochastic transition matrix, I read that its limiting probabilities make up the uniform distribution, but I do not quite understand why.
I have been trying to come up with, and locate, an understandable proof for this. But the proofs I find all gloss over details I don't understand, like proposition 15.5 here (why does it work to just use the [1,...1] vectors?) Could someone point me to (or write) a more simple/detailed proof?
(Though not part of anything I will hand in at school, it is part of a course I'm taking so I guess I'll tag it with homework in either case.)