An asymptotic theory for spectral analysis of random fields (Q1684137)

The authors study a general class of stationary random fields \(X_i\), \(i\in\mathbb{Z}^d\). The asymptotic properties of the discrete Fourier transform of such a process are presented. The White likelihood and parametric spectral density estimator are discussed. The nonparametric spectral density estimator is presented together with estimation of covariance matrices. It is important that the authors do not impose any restriction on the index set \(\Gamma_n\subset\mathbb{Z}^d\), other than the natural requirement \(|\Gamma_n|\rightarrow\infty\). Below we present one statement from the paper on the asymptotic behavior of the discrete Fourier transform. Let \(X_i=g(\varepsilon_{i-s},s\in\mathbb{Z}^d)\), where \(\varepsilon_j\), \(j\in\mathbb{Z}^d\), are i.i.d. random variables, and \(g\) is a measurable function. Let \(W_n=\sum_{j\in\Gamma_n}c_jX_j\) with \(|c_j|\leq 1\) for all \(j\in\Gamma_n\). Suppose that \(\mathbb{E}(W_n^2)\rightarrow\infty\) and \[ \sum_{i\in\mathbb{Z}^d}\|X_i-X_i^*\|_2<\infty, \] where \(X^*_i=g(\varepsilon^*_{i-s},s\in\mathbb{Z}^d)\), \(\varepsilon^*_j=\varepsilon_j\) if \(j\neq 0\) and \(\varepsilon^*_0\) is the independent copy of \(\varepsilon_0\). Then \[ L\Big(\big(Y_n(\theta),Z_n(\theta)\big)/\sqrt{n}, N\big(0,\Sigma_n(\theta)/n\big)\Big)\rightarrow 0, \] where \(Y_n(\theta)=\sum_{j\in\Gamma_n} X_j\cos(j'\theta)\), \(Y_n(\theta)=-\sum_{j\in\Gamma_n} X_j\sin(j'\theta)\), \(\theta\in\mathbb{Z}^d\), \(L\) denotes the Levy distance between distributions and \(N\) is the two-dimensional normal distribution with zero mean vector and covariance matrix \(\Sigma_n(\theta)=\text{cov}\big(Y_n(\theta),Z_n(\theta)\big)\).