 Markov-chain law of large numbers

For a stationary ergodic Markov chain with invariant distribution $\pi$ and an integrable function $h$, the sample mean $n^{-1}{\sum }_{t=1}^{n}h(X_t)$ converges almost surely to $\int hd\pi$. Geometric ergodicity supplies standard sufficient conditions for this conclusion and stronger limit theorems.