Editing Markov chain Monte Carlo (section)

=== Law of Large Numbers for MCMC ===

;Theorem (Ergodic Theorem for MCMC)
If <math>(X_n)</math> has a <math>\sigma</math>-finite invariant measure <math>\pi</math>, then the following two statements are equivalent:
# The Markov chain <math>(X_n)</math> is '''Harris recurrent'''.
# If <math>f, g \in L^1(\pi)</math> with <math>\int g(x) \, d\pi(x) \ne 0</math>, then<math>
\lim_{n \to \infty} \frac{S_n(f)}{S_n(g)} = \frac{\int f(x) \, d\pi(x)}{\int g(x) \, d\pi(x)}.</math>

This theorem provides a fundamental justification for the use of Markov Chain Monte Carlo (MCMC) methods, and it serves as the counterpart of the [[Law of Large Numbers]] (LLN) in classical Monte Carlo. 

An important aspect of this result is that <math>\pi</math> does not need to be a probability measure. Therefore, there can be some type of strong stability even if the chain is null recurrent. Moreover, the Markov chain can be started from arbitrary state.

If <math>\pi</math> is a probability measure, we can let <math>g \equiv 1</math> and get
:<math>
\lim_{n \to \infty} S_n(f) = \int f(x) \,  d\pi(x).
</math>
This is the Ergodic Theorem that we are more familiar with.