On a boundary of the central limit theorem for strictly stationary, reversible Markov chains

Consider the class of (functions of) strictly stationary Markov chains in which (i) the second moments are finite and (ii) absolute regularity (-mixing) is satisfied with exponential mixing rate.For (functions of) Markov chains in that class that are also reversible, the central limit theorem holds, as a well known by-product of results of Roberts, Rosenthal, and Tweedie in two papers in 1997 and 2001 involving reversible Markov chains.In contrast, for (functions of) Markov chains in that class that are not reversible, the central limit theorem may fail to hold, as is known from counterexamples, including ones with arbitrarily fast mixing rate (for absolute regularity).Here it will be shown that for Markov chains in that class that are reversible, the "borderline" class of mixing rates (for absolute regularity) for the central limit theorem is in fact exponential.That will be shown here with a class of counterexamples -strictly stationary, countable-state Markov chains that are reversible, have finite second moments, and satisfy absolute regularity with mixing rates that can be arbitrarily close to (but not quite) exponential, but fail to satisfy the central limit theorem.