Geometric ergodicity often implies a central limit theorem (CLT).
Let g be any 
-integrable function,
where is the stationary distribution of a Harris recurrent chain,
and let
be the sample mean of a run of length n starting at 
, which
may be chosen arbitrarily. Then the strong law of large numbers
(Meyn and Tweedie, 1993, Theorem 17.1.7) implies 
with probability one.
If the function g satisfies a Lyapunov condition
for some 
then the CLT also holds
where
where 
(Chan and Geyer, 1994, Theorem 2). A
different CLT is given by Meyn and Tweedie (1993, Theorem 17.5.4).
They replace the
Lyapunov condition (1.24) by a condition that relates g
to the function V in the geometric drift condition. If 
,
then the CLT (1.25) holds with 
given by (1.26).
When the chain is Harris recurrent, the CLT and the SLLN hold for every starting position, indeed every starting distribution, if they hold for one starting position (Meyn and Tweedie, 1993, Theorem 17.1.6). The idea that one must run the chain until it `reaches equilibrium' before starting to sample, something that is, strictly speaking, impossible, is not required by the asymptotics.
