Gaussian limit fields for the integrated periodogram

Abstract

Functionals of a two-parameter integrated periodogram have been used for detecting a change in the spectral distribution of a stationary sequence. The bases for these results are functional central limit theorems for the integrated periodogram with a Gaussian limit field. We prove functional central limit theorems for a general linear sequence having a finite fourth moment which is shown to be the optimal moment condition. Our approach is via an approximation of the integrated periodogram by a finite linear combination of sample autocovariances. This gives special insight into the structure of the Gaussian limit field.