The Annals of Probability

Central limit theorem for Fourier transforms of stationary processes

Magda Peligrad and Wei Biao Wu

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We consider asymptotic behavior of Fourier transforms of stationary ergodic sequences with finite second moments. We establish a central limit theorem (CLT) for almost all frequencies and also an annealed CLT. The theorems hold for all regular sequences. Our results shed new light on the foundation of spectral analysis and on the asymptotic distribution of periodogram, and it provides a nice blend of harmonic analysis, theory of stationary processes and theory of martingales.

Article information

Ann. Probab., Volume 38, Number 5 (2010), 2009-2022.

First available in Project Euclid: 17 August 2010

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles

Fourier transform spectral analysis martingale central limit theorem stationary process


Peligrad, Magda; Wu, Wei Biao. Central limit theorem for Fourier transforms of stationary processes. Ann. Probab. 38 (2010), no. 5, 2009--2022. doi:10.1214/10-AOP530.

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