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