The periodogram of an i.i.d. sequence

Periodogram ordinates of a Gaussian white-noise computed at Fourier frequen cies are well known to form an i.i.d. sequence. This is no longer true in t he non-Gaussian case. In this paper, we develop a full theory for weighted sums of non-linear functionals of the periodogram of an i.i.d sequence. We prove that these sums are asymptotically Gaussian under conditions very clo se to those which are sufficient in the Gaussian case, and that the asympto tic variance differs from the Gaussian case by a term proportional to the f ourth cumulant of the white noise. An important consequence is a functional central limit theorem for the spectral empirical measure. The technique us ed to obtain these results is based on the theory of Edgeworth expansions f or triangular arrays. (C) 2001 Elsevier Science B.V. All rights reserved.