Vol. 96, No. 2, 1981

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The spectral density of a strongly mixing stationary Gaussian process

Eric Hayashi

Vol. 96 (1981), No. 2, 343–359

Let w be a nonnegative integrable weight function on the real line R such that (log w)(1 + x2) is also integrable. Let FT and PT denote, respectively, the closed linear spans in L2(R,wdx) of {eiax : a T} and {eiax : a T}. Let 𝜃(T) denote the angle between P0 and FT. 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 P0 and FT is introduced: let 𝜃 (T) denote the angle between PT (PT F0) and F0 (PT F0). A complete structural characterization 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.

Mathematical Subject Classification 2000
Primary: 46E30
Secondary: 60G10
Received: 22 May 1979
Published: 1 October 1981
Eric Hayashi