This theorem is in my book, let me just say that it is for discrete-time Markov chains, that are time-homogeneous. Ergodic is defined in the book as being positive recurrent and aperiodic.
The problem is that this theorem is not proved. It just shown that if the probabilities exist, then they must be calculated like this. But existence is not proved. Is existence hard to prove? What kind of mathematical theory or skills does one need to prove this?
I thought that maybe linear algebra was enough, because when we have finite states we are using matrices. However this theorem also holds if we have a countable number of states, provided they all communicate and the class is aperiodic.
PS: Wikipedia does also not have a proof for this.: http://en.wikipedia.org/wiki/Markov_chain#Steady-state_analysis_and_limiting_distributions
So to sum up. Is it hard to prove existence? If it is hard to prove existence, what kind of theory is needed to prove it?
PS: It is also not shown that they really are independent of the starting-state.
This theorem is proved in most textbooks on Markov chains (yours would seem to be an exception). For example, you could see Durrett's Essentials of Stochastic Processes. The proof is not quite trivial but not excessively difficult either. It takes perhaps a page or two once the necessary background is in place.
In the case of a finite-state Markov chain, you can prove the ergodic theorem using linear algebra. The essential piece is the Perron-Frobenius Theorem which also is not quite trivial.