An asymptotic theory for spectral analysis of random fields
type: Item
instance of: scholarly article
title: An asymptotic theory for spectral analysis of random fields
zbMATH Open document ID: 1383.62209
DOI: 10.1214/17-EJS1326
author: Soudeep Deb
author: M. Pourahmadi
author: Wei Biao Wu
published in: Electronic Journal of Statistics
publication date: 2017-12-08T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ejs/1510563632
review text: 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 \(\Sigma_n(\theta)=\text{cov}\big(Y_n(\theta),Z_n(\theta)\big)\).
reviewed by: Jonas Šiaulys
Mathematics Subject Classification ID: 62M15
Mathematics Subject Classification ID: 62M40
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62G20
zbMATH DE Number: 6816617
zbMATH Keywords: time series
zbMATH Keywords: random fields
zbMATH Keywords: discrete Fourier transform
zbMATH Keywords: periodogram
zbMATH Keywords: irregular spaced data
zbMATH Keywords: parametric estimate
zbMATH Keywords: nonparametric estimate
zbMATH Keywords: spectral density
zbMATH Keywords: covariance matrix
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6816617