On the integral of the squared periodogram

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 ∫−ππη(λ)φ(I(λ)) dλ is equivalent to that of the discrete version of the integral given by (2π/n)∑i=1n−1η(λi)φ(I(λi)), where λi are the Fourier frequencies and φ and η are suitable possibly non-linear functions. In this paper, we prove that this asymptotic equivalence is not true when φ is a non-linear function. We derive the exact finite sample variance of ∫−ππI2(λ) dλ when {Xt} is Gaussian white noise and show that it is asymptotically different from that of (2π/n)∑i=1n−1I2(λi). The asymptotic distribution of ∫−ππI2(λ) dλ is also obtained in this case. The result is then extended to obtain the limiting distribution of ∫−ππf−2(λ)I2(λ) dλ 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.