On the integral of the squared periodogram
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Let X1, X2.,., Xn 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 -()(I())d is equivalent to that of the discrete version of the integral given by (2/n)i=1n-1(i)(I(i)), where i are the Fourier frequencies and and are suitable possibly non-linear functions. In this paper, we prove that this asymptotic equivalence is not true when is a non-linear function. We derive the exact finite sample variance of -I2()d when (Xt) is Gaussian white noise and show that it is asymptotically different from that of (2/n)i=1n-1I2(i). The asymptotic distribution of -I2()d is also obtained in this case. The result is then extended to obtain the limiting distribution of -f-2()I2()d when (Xt) 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. 2000 Elsevier Science B.V.
