Open Access
February 2019 Central limit theorem for Fourier transform and periodogram of random fields
Magda Peligrad, Na Zhang
Bernoulli 25(1): 499-520 (February 2019). DOI: 10.3150/17-BEJ995


In this paper, we show that the limiting distribution of the real and the imaginary part of the Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field’s spectral density. The dependence structure of the random field is general and we do not impose any restrictions on the speed of convergence to zero of the covariances, or smoothness of the spectral density. The only condition required is that the variables are adapted to a commuting filtration and are regular in some sense. The results go beyond the Bernoulli fields and apply to both short range and long range dependence. They can be easily applied to derive the asymptotic behavior of the periodogram associated to the random field. The method of proof is based on new probabilistic methods involving martingale approximations and also on borrowed and new tools from harmonic analysis. Several examples to linear, Volterra and Gaussian random fields will be presented.


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Magda Peligrad. Na Zhang. "Central limit theorem for Fourier transform and periodogram of random fields." Bernoulli 25 (1) 499 - 520, February 2019.


Received: 1 April 2017; Revised: 1 October 2017; Published: February 2019
First available in Project Euclid: 12 December 2018

zbMATH: 07007215
MathSciNet: MR3892327
Digital Object Identifier: 10.3150/17-BEJ995

Keywords: central limit theorem , Fourier transform , Martingale approximation , Random field , Spectral density

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 1 • February 2019
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