Quenched central limit theorems for sums of stationary processes

It is shown that the existence of an L1 martingale-coboundary decomposition does not imply the quenched version of the Central Limit Theorem. In another result, it is shown that a condition proposed by Hannan does imply quenched convergence for a centered version of the sum while a condition proposed by Heyde does not imply quenched convergence.