An asymptotic theory for spectral analysis of random fields

Abstract

For a general class of stationary random fields we study asymptotic properties of the discrete Fourier transform (DFT), periodogram, parametric and nonparametric spectral density estimators under an easily verifiable short-range dependence condition expressed in terms of functional dependence measures. We allow irregularly spaced data which is indexed by a subset $\Gamma $ of $\mathbb{Z}^{d}$. Our asymptotic theory requires minimal restriction on the index set $\Gamma $. Asymptotic normality is derived for kernel spectral density estimators and the Whittle estimator of a parameterized spectral density function. We also develop asymptotic results for a covariance matrix estimate.