In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theory, but the quantity in the role taken by the variance in the classic CLT has a more complicated definition. See also the general form of Bienaymé's identity.

Suppose that:

Now let

Then as ${\textstyle n\to \infty ,}$ we have

where the decorated arrow indicates convergence in distribution.

The Markov chain central limit theorem can be guaranteed for functionals of general state space Markov chains under certain conditions. In particular, this can be done with a focus on Monte Carlo settings. An example of the application in a MCMC (Markov Chain Monte Carlo) setting is the following:

Consider a simple hard spheres model on a grid. Suppose ${\displaystyle X=\{1,\ldots ,n_{1}\}\times \{1,\ldots ,n_{2}\}\subseteq Z^{2}}$. A proper configuration on ${\displaystyle X}$ consists of coloring each point either black or white in such a way that no two adjacent points are white. Let ${\displaystyle \chi }$ denote the set of all proper configurations on ${\displaystyle X}$, ${\displaystyle N_{\chi }(n_{1},n_{2})}$ be the total number of proper configurations and π be the uniform distribution on ${\displaystyle \chi }$ so that each proper configuration is equally likely. Suppose our goal is to calculate the typical number of white points in a proper configuration; that is, if ${\displaystyle W(x)}$ is the number of white points in ${\displaystyle x\in \chi }$ then we want the value of

${\displaystyle E_{\pi }W=\sum _{x\in \chi }{\frac {W(x)}{N_{\chi }{\bigl (}n_{1},n_{2}{\bigr )}}}}$