Central limit theorem for Fourier transforms of stationary processes

2009/10/31 by Magda Peligrad, Wei Biao Wu · 1 citation
Mathematics · #math.PR

paper · pdf · doi:10.1214/10-aop530

published as Annals of Probability 2010, Vol. 38, No. 5, 2009-2022 · Published in at http://dx.doi.org/10.1214/10-AOP530 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

arxiv created 2010/11/05 · arxiv updated 2010/11/08

Abstract

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.

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