On the integral of the squared periodogram

Abstract

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 ∫ − π π η(λ)φ(I(λ)) dλ is equivalent to that of the discrete version of the integral given by (2 π/n) ∑ i=1 n−1 η(λ i)φ(I(λ i)) , where λ i 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 ∫ − π π I 2(λ) dλ when { X t } is Gaussian white noise and show that it is asymptotically different from that of (2 π/n) ∑ i=1 n−1 I 2(λ i) . The asymptotic distribution of ∫ − π π I 2(λ) dλ is also obtained in this case. The result is then extended to obtain the limiting distribution of ∫ − π π f −2(λ)I 2(λ) dλ 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.