We assume the Markov chain to be countable state space, timehomogeneous. Does it necessarily have a stationary distribution? I found a paper on arXiv.org (http://arxiv.org/abs/math/0610707) that proves that for every continuous transformation from the standard infinite dimensional (the convex hull of the standard bases of $\mathbb{R}^\infty$) simplex to itself has a fixed point. So I guess it necessarily has but when I look for such a theorem on books, I cannot find one. Thank you for your help!
