  Kipnis and Varadhan showed that for an additive functional, $S_n$ say, of a
reversible Markov chain the condition $E(S_n^{2})/n \to \kappa \in (0,\infty)$
implies the convergence of the conditional distribution of
$S_n/\sqrt{E(S_n^{2}})$, given the starting point, to the standard normal
distribution. We revisit this question under the weaker condition, $E(S_n^{2})
= n\ell(n)$, where $\ell$ is a slowly varying function. It is shown by example
that the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$ need not converge
to the standard normal distribution in this case; and sufficient conditions for
convergence to a (possibly non-standard) normal distribution are developed.
