Title: Asymptotic normality of integrated periodogram operators
Authors: Daniel Rademacher - TU Braunschweig (Germany) [presenting]
Jens-Peter Kreiss - Technische Universitaet Braunschweig (Germany)
Efstathios Paparoditis - University of Cyprus (Cyprus)
Abstract: Consider a strictly stationary functional process. A key element of a frequency domain framework for drawing statistical inference on the second-order structure of the process is the spectral density operator, which generalises the notion of a spectral density matrix to the functional setting. As an integral operator, the spectral density operator is fully determined by its corresponding kernel, which can be estimated by a smoothed version of the periodogram kernel (the functional analogue to the periodogram matrix). More generally, many interesting quantities of the process such as autocovariance operators, or the spectral distribution operator can be represented as a weighted integral of the spectral density kernel. Estimators for such a quantity are obtained by exchanging the spectral density kernel with the periodogram kernel. Thus, the class of integrated periodogram operators covers many familiar statistics, including empirical autocovariance and smoothed periodogram operators. We show that any finite collection of such estimators converges to a collection of jointly complex normal distributed operators. As a side-result, we obtain the joint asymptotic normality for the empirical autocovariance operators. These results do not depend on any structural modelling assumptions, but only on functional versions of cumulant mixing conditions.