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

Suppose that:

• the sequence ${\textstyle X_{1},X_{2},X_{3},\ldots }$ of random variables is a Markov chain that has a stationary probability distribution; and
• the initial distribution of the process, i.e. the distribution of ${\textstyle X_{1}}$, is the stationary distribution, so that ${\textstyle X_{1},X_{2},X_{3},\ldots }$ are identically distributed. In the classic central limit theorem these random variables would be assumed to be independent, but here we have only the weaker assumption that the process has the Markov property; and
• ${\textstyle g}$ is some (measurable) function for which ${\textstyle \operatorname {var} (g(X_{1}))<+\infty .}$

Now let

${\begin{aligned}\mu &=\operatorname {E} (g(X_{1})),\\\sigma ^{2}&=\operatorname {var} (g(X_{1}))+2\sum _{k=1}^{\infty }\operatorname {cov} (X_{1},X_{1+k}),\\{\widehat {\mu }}_{n}&={\frac {1}{n}}\sum _{k=1}^{n}g(X_{k}).\end{aligned}}$

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

${\widehat {\mu }}_{n}\approx \operatorname {Normal} \left(\mu ,{\frac {\sigma ^{2}}{n}}\right),$

or more precisely, for every (measurable) set ${\textstyle A}$ of real numbers,

$\lim _{n\to \infty }\Pr \left({\frac {{\widehat {\mu }}_{n}-\mu }{\sigma /{\sqrt {n}}}}\in A\right)=\int _{A}\varphi (x)\,dx$

where

$\varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}$

is the probability density function of the standard (zero mean, unit variance) normal distribution.