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April 2004 Martingale approximations for sums of stationary processes
Wei Biao Wu, Michael Woodroofe
Ann. Probab. 32(2): 1674-1690 (April 2004). DOI: 10.1214/009117904000000351

Abstract

Approximations to sums of stationary and ergodic sequences by martingales are investigated. Necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained. It is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating martingales satisfy a Lindeberg–Feller condition. Using the explicit construction of the approximating martingales, a central limit theorem is derived for the sample means of linear processes. The conditions are not sufficient for the functional version of the central limit theorem. This is shown by an example, and a slightly stronger sufficient condition is given.

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Wei Biao Wu. Michael Woodroofe. "Martingale approximations for sums of stationary processes." Ann. Probab. 32 (2) 1674 - 1690, April 2004. https://doi.org/10.1214/009117904000000351

Information

Published: April 2004
First available in Project Euclid: 18 May 2004

zbMATH: 1057.60022
MathSciNet: MR2060314
Digital Object Identifier: 10.1214/009117904000000351

Subjects:
Primary: 60F05 , 60F17 , 60G42
Secondary: 60J10

Keywords: central limit theorem , invariance principle , linear process , Markov chain , martingale , Poisson equation , stationary process

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 2 • April 2004
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