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.
we have[4] 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.
A proper configuration on
consists of coloring each point either black or white in such a way that no two adjacent points are white.
denote the set of all proper configurations on
be the total number of proper configurations and π be the uniform distribution on
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
is the number of white points in
π
are even moderately large then we will have to resort to an approximation to
Consider the following Markov chain on
is an arbitrary proper configuration.
Randomly choose a point
and independently draw
and all of the adjacent points are black then color
white leaving all other points alone.
black and leave all other points alone.
Call the resulting configuration
Continuing in this fashion yields a Harris ergodic Markov chain
as its invariant distribution.
It is now a simple matter to estimate
is finite (albeit potentially large) it is well known that
will converge exponentially fast to
which implies that a CLT holds for
Not taking into account the additional terms in the variance which stem from correlations (e.g. serial correlations in markov chain monte carlo simulations) can result in the problem of pseudoreplication when computing e.g. the confidence intervals for the sample mean.