Markov chain central limit theorem


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 of probability theory, but the quantity in the role taken by the variance in the classic CLT has a more complicated definition.

Statement

Suppose that:
Now let
Then as we have
or more precisely,
where the decorated arrow indicates convergence in distribution.

Monte Carlo Setting

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 setting is the following:
Consider a simple hard-shell model. Suppose X = × ⊆ Z 2. A proper configuration on X consists of coloring each point either black or white in such a way that no two adjacent points are white. Let X denote the set of all proper configurations on X, N X be the total number of proper configurations and π be the uniform distribution on X 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 W is the number of white points in x ∈ X then we want the value of
If n1 and n2 are even moderately large then we will have to resort to an approximation to E π W. Consider the following Markov chain on X. Fix p ∈ and set X 0 = x 0 where x 0 ∈ X is an arbitrary proper configuration. Randomly choose a point ∈ X and independently draw U ∼ Uniform. If u ≤ p and all of the adjacent points are black then color white leaving all other points alone. Otherwise, color black and leave all other points alone. Call the resulting configuration X 1. Continuing in this fashion yields a Harris ergodic Markov chain having π as its invariant distribution. It is now a simple matter to estimate E π W with w̄ n. Also, since X is finite it is well known that X will converge exponentially fast to π which implies that a CLT holds for w̄ n.