Asymptotic normality of the discrete Fourier transform of long memory time series
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.
scientific article; zbMATH DE number 706904