An asymptotic Wiener-Itô representation for the low frequency ordinates of the periodogram of a long memory time series

We consider a general long memory time series, assumed stationary and linear, but not necessarily Gaussian or generated by a finite-parameter model. For such a process, we derive the asymptotic joint distribution of the normalized periodogram at a fixed, finite collection of Fourier frequencies. The limiting distribution is represented in terms of Wiener-Itô integrals, and, for a single periodogram ordinate, it is an unequally weighted linear combination of independent [chi]21 random variables. This result was previously known only in the Gaussian case. Our theorem may be useful for generalizing, beyond the Gaussian case, the applicability of a semiparametric method of estimating the long memory parameter based on log-periodogram regression.