Tapering Promotes Propriety for Fourier Transforms of Real-Valued Time Series

We examine Fourier transforms of real-valued stationary time series from the point of view of the statistical propriety. Processes with a large dynamic range spectrum have transforms that are very significantly improper for some frequencies; the real and imaginary parts can be highly correlated, and the periodogram will not have the standard chi-square distribution at these frequencies, nor have two degrees of freedom. Use of a taper reduces impropriety to just frequencies close to zero and Nyquist only, and frequency ranges where the propriety breaks down can be quite accurately and easily predicted by half the autocorrelation width of <inline-formula><tex-math notation="LaTeX"> $|H*H(2f)|,$</tex-math></inline-formula> denoted by <inline-formula><tex-math notation="LaTeX">$c,$</tex-math> </inline-formula> where <inline-formula><tex-math notation="LaTeX">$H(f)$</tex-math></inline-formula> is the Fourier transform of the taper and <inline-formula><tex-math notation="LaTeX">$*$</tex-math></inline-formula> denotes convolution. For vector time series we derive an improved distributional approximation for minus twice the log of the generalized likelihood ratio test statistic for testing for propriety of the Fourier transform at any frequency, and compare frequency range cutoffs for propriety determined by the hypothesis test with those determined by <inline-formula><tex-math notation="LaTeX">$c.$</tex-math></inline-formula>