On the exactness of the Wu–Woodroofe approximation

Let (Xi)(Xi) be a stationary process adapted to a filtration (Fi)(Fi), E(Xi)=0E(Xi)=0, E(Xi2)<∞; by Sn=∑i=0n−1Xi we denote the partial sums and σn2=‖Sn‖22. Wu and Woodroofe [Wei Biao Wu, M. Woodroofe, Martingale approximation for sums of stationary processes, Ann. Probab. 32 (2004) 1674–1690] have shown that if ‖E(Sn∣F0)‖2=o(σn)‖E(Sn∣F0)‖2=o(σn) then there exists an array of row-wise stationary martingale difference sequences approximating the partial sums SnSn. If ∑n=1∞‖E(Sn∣F0)‖2n3/2<∞ then by [M. Maxwell, M. Woodroofe, Central limit theorems for additive functionals of Markov chains, Ann. Probab. 28 (2000) 713–724] there exists a stationary martingale difference sequence approximating the partial sums SnSn, and the central limit theorem holds. We will show that the process (Xi)(Xi) can be found so that ‖E(Sn∣F0)‖2=O(nlog1/2n), σn2/n→ constant but the central limit theorem does not hold. The linear growth of the variances σn2 is a substantial source of complexity of the construction.