Central limit theorem for Fourier transform and periodogram of random elds
نویسنده
چکیده
In this paper we show that the limiting distribution of the real and the imaginary part of the Fourier transform of a stationary random eld is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the elds spectral density. The dependence structure of the random eld 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 ltration and are regular in some sense. The results go beyond the Bernoulli elds 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 eld. 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 elds will be presented. MSC: Primary: 60F05, 60G10, 60G12, Secondary: 42B05 Keywords: random eld; central limit theorem; Fourier transform; spectral density; martingale approximation. 1 Introduction The discrete Fourier transform, de ned as
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