Where can I find a proof of the erogidc theorem for Markov chains that doesn't use Birkhoff?
The theorem states the following :
Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible and positively recurrent Markov chain in a countable state space $E$ with invariant measure $\pi$. Then for every function $f \in L^{1}(E,\pi)$
$$\frac{1}{N}\sum_{n=0}^{N-1}f(X_n) \stackrel{N\to \infty}{\longrightarrow} \int_E fd\pi=\sum_{k\in E}\pi_k f(k) \quad\mathrm{ a.e.}$$
You can find an elementary proof in Durrett's Essentials of Stochastic Processes, at the end of Chapter 1.