The Annals of Probability

Martingale approximations for sums of stationary processes

Wei Biao Wu and Michael Woodroofe

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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.

Article information

Ann. Probab., Volume 32, Number 2 (2004), 1674-1690.

First available in Project Euclid: 18 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60G42: Martingales with discrete parameter 60F05: Central limit and other weak theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Central limit theorem invariance principle linear process Markov chain martingale Poisson equation stationary process


Wu, Wei Biao; Woodroofe, Michael. Martingale approximations for sums of stationary processes. Ann. Probab. 32 (2004), no. 2, 1674--1690. doi:10.1214/009117904000000351.

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