On the Covariance of the Periodogram

The paper discusses the covariance of the periodogram from a zero mean fourth order stationary stochastic process. The fourth order cumulant term appearing in the covariance is shown to be a convolution between the fourth order cumulant spectrum and a bounded approximate identity, and this gives precise results about its asymptotic behaviour. The covariance is also studied both pointwise and as a measure of two variables. This leads to necessary and sufficient conditions for mean square consistency of estimates of the spectral moments and related parameters.