Asymptotic normality of the discrete Fourier transform of long memory time series
type: Item
instance of: scholarly article
title: Asymptotic normality of the discrete Fourier transform of long memory time series
zbMATH Open document ID: 0805.62088
DOI: 10.1016/0167-7152(94)00023-9
author: Phm Dinh Tuan
author: Dominique Guegan
published in: Statistics \& Probability Letters
publication date: 1995-01-09T00:00:00Z
review text: One of the central results in time-series analysis, particularly in the context of spectrum estimation, is that the values of the discrete Fourier transform at distinct frequencies are asymptotically independent and normally distributed. This result holds under quite general regular assumptions which however implies that the underlying process is of short-range memory. Recently, there has been considerable interest in long-range memory processes and one may ask if the above result continues to hold in such situations. This problem has been addressed in a paper of \textit{Y. Yajima} [J. Time Ser. Anal. 10, No. 4, 375-383 (1989; Zbl 0692.60024)]. However, Yajima's result is not quite complete since the frequencies where the Fourier transforms are considered are not allowed to vary with the sample size \(T\). In many situations (such as spectrum estimation or testing for a long-memory model or estimating the characteristic exponent of the spectral density near the origin of such model), one would need to consider a set of frequencies tending to some given frequency (zero in particular) as the sample size goes to infinity. The purpose of this work is to extend Yajima's result to cover this case. Instead of trying to adapt Yajima's proofs, which we have some difficulties to follow because of misprints, we have derived our own alternative proof for this general situation.
Mathematics Subject Classification ID: 62M15
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 706904
zbMATH Keywords: spectral density
zbMATH Keywords: time-series
zbMATH Keywords: spectrum estimation
zbMATH Keywords: discrete Fourier transform
zbMATH Keywords: short-range memory
zbMATH Keywords: long-range memory processes
MaRDI profile type: Publication
full work available at URL: 0167-7152(94)00023-9
OpenAlex ID: W2058639226
cites work: On large-sample estimation for the mean of a stationary random sequence
cites work: Q3947020
cites work: The central limit theorem for time series regression
cites work: A central limit theorem for stationary processes and the parameter estimation of linear processes
cites work: On estimation of a regression model with long-memory stationary errors
cites work: A CENTRAL LIMIT THEOREM OF FOURIER TRANSFORMS OF STRONGLY DEPENDENT STATIONARY PROCESSES
description: scientific article; zbMATH DE number 706904