The spectral density of a strongly mixing stationary Gaussian process
Eric Hayashi · Pacific Journal of Mathematics · 1981
Let w be a nonnegative integrable weight function on the real line R such that (log w)l(l J rX 2 ) is also integrable.Let F τ and P τ denote, respectively, the closed linear spans in L 2 (R, wax) of {e iax : a^T} and {e iax : a^ T}.Let Θ(T) denote the angle between P o and F τ .The problem considered here is that of describing those weights w for which θ(T)->π!2 as T tends to infinity (such weights arise as the spectral densities of strongly mixing stationary Caussian processes).Some necessary conditions on w are given for Θ(T) ->π/2, and a construction is given to show that w may have arbitrarily wild oscillatory discontinuities even if θ(T)->π/2.Another measure of the interdependence of P o and F τ is introduced: let Θ*{T) denote the angle between P τ θ(P τ nF 0 ) and F 0 Q(P T i]F Q ).A complete structural char acterization is given of those weights w for which both Θ(T) and Θ*(T) tend to π/2.Moreover, it is shown that if either Θ{T) or Θ*(T) is eventually positive and the other tends to π/2, then they both do.