On the exactness of the Wu-Woodroofe approximation

Jana Klicnarová and Dalibor Volný

Stochastic Processes and their Applications, 2009, vol. 119, issue 7, 2158-2165

Abstract: Let (Xi) be a stationary process adapted to a filtration , E(Xi)=0, ; by we denote the partial sums and . 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 then there exists an array of row-wise stationary martingale difference sequences approximating the partial sums Sn. If 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 Sn, and the central limit theorem holds. We will show that the process (Xi) can be found so that , constant but the central limit theorem does not hold. The linear growth of the variances is a substantial source of complexity of the construction.

Keywords: Martingale; approximation; Central; limit; theorem; Stationary; process (search for similar items in EconPapers)
Date: 2009
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