Let X(t) (t=O, ± 1, ... ) be a zero mean, r vector-valued, strictly stationary time series satisfying a particular assumption about the near-independence of widely separated values. Given the values X(t) (t= 0,1, ... , T -1), we construct the statistics: I1.k(i\.) (-00 < i\. < 00), the matrix of second-order periodograms, F~1 (i\.), the matrix of sample spectral measures, f~l(i\.), the matrix of sample spectral densities and c<f~(u) (u= 0, ± 1, ... ), the matrix of sample covariances. In the

more »

... er expressions are derived for the first-and second-order moments and the asymptotic distributions of 11.k(i\.), F~i1(i\.), f~l(i\.) and c~1(u). Our purpose is to determine the form of these moments and to indicate the appearance of the Wishart distribution as an exact limiting distribution for f<f.k(i\.). It has previously been suggested as an approximation. 24 Biom. 56 P. Guttorp and D. Brillinger (eds.), Selected Works ofDavid Brillinger, Selected Works in Probability 179 and Statistics, DOl 10.1007/978-1-4614-1344-8_12,