On relations between prediction error covariance of univariate and multivariate processes (Q1210125)

Let \(\{x_ t\}\) be a zero-mean, purely deterministic stationary process, with spectral density \(f_ x(\lambda)\), and \(\{X_ t\}\) a \(q\)-variate process defined by \(X_ t=(x_{qt},x_{qt+1},\dots,x_{qt+q-1})'\). Then it is proved that \(\{X_ t\}\) is a stationary process with the \(q\times q\) spectral density matrix \(f_ X(\lambda)\) given by \(f_ X(\lambda)=U(\lambda)D(\lambda)U^*(\lambda)\), where \(D\) is a diagonal matrix and \(U(\lambda)\) is a unitary matrix. Also, the paper obtains closed-form and applicable formulae for the interpolators and their errors for a stretch of missing values of univariate processes.