On the exactness of the Wu–Woodroofe approximation

Let ( X i ) be a stationary process adapted to a filtration ( F i ) , E ( X i ) = 0 , E ( X i 2 ) < ∞ ; by S n = ∑ i = 0 n − 1 X i we denote the partial sums and σ n 2 = ‖ S n ‖ 2 2 . 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 ( S n ∣ F 0 ) ‖ 2 = o ( σ n ) then there exists an array of row-wise stationary martingale difference sequences approximating the partial sums S n . If ∑ n = 1 ∞ ‖ E ( S n ∣ F 0 ) ‖ 2 n 3 / 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 S n , and the central limit theorem holds. We will show that the process ( X i ) can be found so that ‖ E ( S n ∣ F 0 ) ‖ 2 = O ( n log 1 / 2 n ) , σ n 2 / n → constant but the central limit theorem does not hold. The linear growth of the variances σ n 2 is a substantial source of complexity of the construction.