Central limit theorem for Fourier transforms of stationary processes

Details Central limit theorem for Fourier transforms of stationary processes Central limit theorem for Fourier transforms of stationary processes

2009-10-19

arxiv.org/abs/0910.3451v3

Annals of Probability 2010, Vol. 38, No. 5, 2009-2022

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP530 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP530

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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