Generalized autocovariance matrices for multivariate time series

The paper treats the modeling of stationary multivariate stochastic processes via frequency domain, and extends the notion of generalized autocovariance function, given by Proietti and Luati (2015) for univariate time series, to the multivariate setting. The generalized autocovariance matrices are defined for stationary multivariate stochastic processes as the Fourier transform of the power transformation of the spectral density matrix. Then we prove the consistency and derive the asymptotic distribution of frequency domain non-parametric estimators of the generalized autocovariance matrices, based on the power transformation of the periodogram matrix. Generalized autocovariance matrices are used to construct white noise hypothesis testing, to discriminate stochastic processes, and to introduce a generalized Yule-Walker estimator for the spectrum. A so-called lambda-squared distance between two multivariate stochastic processes is also defined by using their generalized autocovariance matrices, and it serves for clustering time series and estimation by feature matching. Another use is in discriminant analysis.