# How to prove ergodic property from aperiodicity and positive recurrence

How to prove that in case of an irreducible, aperiodic and positive recurrent Markov Chain time average along sample paths is equal to the ensemble average ? i.e.

$$\lim_{n\to \infty }\frac{1}{n}\sum _ {i=1} ^{n} X_{i} \rightarrow E[X_k]$$

where $E[X_k]$ is calculated from the steady state probability distribution.

Now each $X_i$'s are dependent because it is a Markov Chain. So, not possible to apply law of large numbers.

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