On the integral of the squared periodogram

Let X-1, X-2,..., X-n be a sample from a stationary Gaussian time series and let I() be the sample periodogram. Some researchers have either proved heuristically or claimed that under general conditions, the asymptotic behaviour of integral(-pi)(pi) eta(lambda)phi(I(lambda)) d lambda is equivalent to that of the discrete version of the integral given by (2 pi/n) Sigma(i=1)(n-1) eta(lambda(i))phi(I(lambda(i))), where lambda(i) are the Fourier frequencies and phi and eta are suitable possibly non-linear functions. In this paper, we prove that this asymptotic equivalence is not true when phi is a non-linear function. We derive the exact finite sample variance of integral(-pi)(pi) I-2(lambda)d lambda when {X-t} is Gaussian white noise and show that it is asymptotically different from that of (2 pi/n) Sigma(i=1)(n-1) I-2(lambda(i)). The asymptotic distribution of integral(-pi)(pi) I-2 (lambda) d lambda is also obtained in this case. The result is then extended to obtain the limiting distribution of integral(-pi)(pi) f(-2) (lambda)I-2(lambda) d lambda when {X-t} is a stationary Gaussian series with spectral density f(.). From these results, the limiting distribution of the integral version of a goodness-of-fit statistic proposed in the literature is obtained. (C) 2000 Elsevier Science B.V. All rights reserved. MSG: primary 62E20; secondary 62F03.