Frequency Domain Tests Of Multivariate Gaussianity And Linearity

A stationary multivariate time series {Xt} is defined as linear if it can be written in the form Xt = ∑∞j=−∞Ajet−j where Aj are square matrices and et are independent and identically distributed random vectors. If the et} are normally distributed, then {Xtis a multivariate Gaussian linear process. This paper is concerned with the testing of departures of a vector stationary process from multivariate Gaussianity and linearity using the bispectral approach. First the definition and properties of cumulants of random matrices are used to obtain the expressions for the higher‐order cumulant and spectral vectors of a linear vector process as defined above. Then it is shown that linearity of a vector process implies constancy of the modulus square of its normalized higher‐order spectra whereas the component of such a vector process does not necessarily have a linear representation. Finally, statistics for the testing of multivariate Gaussianity and linearity are proposed.