The note is concerned with strongly consistent estimation of the 
Markov chain CLT asymptotic variance in order to construct 
asymptotic confidence intervals.


It is shown that the batch means estimator is consistent under 
the following conditions.

a) mixing condition: the Markov chain is geometrically ergodic.

b) integrability condition: \pi |g|^{2+\delta} < \infty,  
where \pi is the stationary distribution and g is the target function

c) regulatity condition for number and size of batches, 
see paper for details.


The current result is optimal in tems of mixing and integrability conditions 
and improves over Jones et. al JASA 2006, where stronger integrability 
conditions are required under geometric ergodicity.