I'm looking for a reference which gives sufficient conditions for the variance to be positive in the central limit theorem for Markov chains (cf <https://en.wikipedia.org/wiki/Markov_chain_central_limit_theorem>).
In other words, using the notation on the wikipedia page, I'm looking for a sufficient condition for $\sqrt{n}(\hat\mu\_n-\mu)$ to legitimately be a random variable in the limit. 


In my situation, the Markov chain is simply a random walk on a finite, aperiodic, strongly-connected graph, so a result relating to this simpler situation would be good enough if nothing more general exists. Also, in my situation, actually attempting to estimate $\sigma^2$ via the formula is pretty much out of the question.


I beleive that I can argue that the variance will be positive if there are two directed cycles $v\_0\to v\_2\to\dots\to v\_n=v\_0$ and $u\_0\to u\_2\to\dots\to u\_m=u\_0$ with ${1\over n}\sum\_{t=1}^n g(v\_t)\neq{1\over m}\sum\_{t=1}^m g(u\_t)$. Intuitively, this is because the chain will always have a chance to move around either of these cycles and these cycles contribute different values to the sum.


I'm not looking for anyone to write down a proof since, if necessary, I can likely prove it myself. However, I would really like to avoid having to spend the necessary pages to do so in a paper I'm writing. It seems very likely that such a result must have appeared in either a paper or book somewhere, but I haven't had any luck in hunting down a reference so far.



Reference request: When is the variance in the central limit theorem for Markov chains positive?
*Fluctuations in Markov processes* by Komorowski, Landim and Olla has more than you could ever hope for in this direction. Also, the Courant lecture notes of Varadhan have, if I recall correctly, a chapter in this direction.


