Bartlett [1] obtained an asymptotic relation connecting the periodogram of a linear process and the periodogram of the residuals with uniform spectrum in the case of one dimensional stationary time series, and using this relation he studied the asymptotic properties of the periodogram. In this paper we obtain some asymptotic relations between the co-periodogram and quadrature periodogram of the residuals in the case of a stationary Gaussian vector process. The covariances of the co-periodogram and quadrature periodogram have also been obtained. Two inequalities connecting the variance and the bandwidth of the cross spectral estimate have been derived.