Ergodic Theorem of Markov Chain

by W. B. Meitei, PhD

If a Markov chain is irreducible, aperiodic, and has a stationary distribution \(\pi( \cdot )\), then for any integrable function \(f\), as \(N \rightarrow \infty\),

\[\frac{1}{N}\sum_{t = 1}^{N}{f\left( X_{t} \right)}\overset{a.s.}{\rightarrow}E_{\pi}\left\lbrack f(X) \right\rbrack\]

where, \(f\) is any bounded function (like log-likelihood, parameter estimates), and \(\ E_{\pi}\lbrack f\rbrack\) is the target posterior expectation.

Consequently, the distribution of \(X_{t}\) converges to \(\pi( \cdot )\) as \(t \rightarrow \infty\), i.e.,

\[\lim_{t \rightarrow \infty}\left\| P\left( X_{t} \in \  \cdot \right) - \pi( \cdot ) \right\|_{TV} = 0\]

where, \(\left\| \ \cdot \ \right\|_{TV}\) denotes the total variation distance. 

Suggested Readings:

1. Robert, C. P., & Casella, G. (2004). Introducing Monte Carlo Methods with R. Springer.
2. Geyer, C. J. (2011). Introduction to Markov Chain Monte Carlo. Handbook of Markov Chain Monte Carlo. CRC Press.
3. Gundersen, G. (2019). Ergodic Markov Chains. 

Suggested Citation: Meitei, W. B. (2026). Ergodic Theorem of Markov Chain. WBM STATS.