Asymptotic properties of spectral estimates of second order
David R. Brillinger · Biometrika · 1969
Let X(t) (t = 0, ± 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: I(T)/XX(γ)(-∞λlt;∞),the matrix of second-order periodograms, FT/XX(λ),the matrix of sample spectral measures, fT/XX(λ), the matrix of sample spectral densities and c(T)/(u) (u = 0,± 1,…), the matrix of sample covariances. In the paper expressions are derived for the first- and second-order moments and the asymptotic distributions of IT/XX(λ), F(T)/XX(λ), f(T)/XX(λ) and c(T)/XX(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(T)/XX(λ). It has previously been suggested as an approximation.