On the exactness of the Wu–Woodroofe approximation

Abstract

Let (X i ) be a stationary process adapted to a filtration (F i ), E(X i ) = 0, E(X 2 i ) < ∞; by S n = n−1 i=0 X i we denote the partial sums and σ 2 n = 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 2 n /n → constant but the central limit theorem does not hold. The linear growth of the variances σ 2 n is a substantial source of complexity of the construction.

Recent progress on the Random Conductance Model

“…The role of these conditions is to control the antisymmetric ("non-reversible") part of the generator by the symmetric one. Another approach goes by imposing decay-rate conditions on time-correlations (e.g., Maxwell and Woodroofe [103], Derriennic and Lin [47], Peligrad and Utev [113], Klicnarová and Volný [88], Volný [133], etc.). However, unlike the reversible situations, it does not seem likely that a single condition will eventually cover all cases of interest.…”

On the optimality of McLeish's conditions for the central limit theorem

“…I would like to thank the referee, who also indicated to me Lemma 2.1 below, and the references [8] and [13].…”

“…In the paper [13], there is an example of a stationary ergodic sequence such that E(S n |F 0 ) 2 = o( √ n/ ln(n)) and the CLT fails, but again the variance does not grow linearly. In [8], there is an example for which E(S n |F 0 ) 2 = o( n/ ln(n)) and n −1 E(S 2 n ) → 1 and the CLT fails. In Remark 2 of [8], the authors ask the following question: does E(S n |F 0 ) 2 = o( √ n/ ln(n)) and n −1 E(S 2 n ) → 1 imply the CLT?…”

“…In [8], there is an example for which E(S n |F 0 ) 2 = o( n/ ln(n)) and n −1 E(S 2 n ) → 1 and the CLT fails. In Remark 2 of [8], the authors ask the following question: does E(S n |F 0 ) 2 = o( √ n/ ln(n)) and n −1 E(S 2 n ) → 1 imply the CLT? Thanks to Theorem 2.2, we are able to give a negative answer to this question: it suffices to take ψ n ≡ 1 and u n = (nL 2,1 (n)) −1 , which implies that E(X n |F 0 ) 2 = o(n −1/2 / ln(n)).…”