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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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