Let X₁,X₂,…,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 is equivalent to that of the discrete version of the integral given by , 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 when {X_t} is Gaussian white noise and show that it is asymptotically different from that of . The asymptotic distribution of is also obtained in this case. The result is then extended to obtain the limiting distribution of 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.