Central limit theorem for Fourier transform and periodogram of random fields

Abstract

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. MSC: Primary: 60F05, 60G10, 60G12, Secondary: 42B05

Functional Central Limit Theorem via Nonstationary Projective Conditions

“…We mention the well-known results, such as the celebrated results by Gordin [12], Heyde [18], Maxwell and Woodroofe [21] and the more recent results by Peligrad and Utev [30], Zhao and Woodroofe [43], Gordin and Peligrad [14], among many others. In the context of random fields, the theory of martingale approximation has been developed in the last decade, with several results by Gordin [13], Volný and Wang [41], Cuny et al [3], El Machkouri and Giraudo [10], Peligrad and Zhang [32,33,34], Giraudo [11] and Volný [39,40]. Due to these results we know now necessary and sufficient conditions for various types of martingale approximations which lead to a variety of maximal inequalities and limit theorems.…”