A note on the martingale approximation method in proving the central limit theorem for stationary random sequences

Under appropriate assumptions the martingale approximation method allows to reduce the study of the asymptotic behavior of sums of random variables forming a stationary random sequence to the analogous problem about the sums of stationary martingale differences. In his early paper on the martingale method the author proposed certain sufficient conditions for the central limit theorem to hold. It is shown in the present note that these conditions, at least in one particular case, can be essentially relaxed. In the context of the central limit theorem for Markov chains an analogous observation was made in a recent paper by H. Holzmann and the author.