Estimation of the Cross-Spectrum of a Stationary Bivariate Gaussian Process from its Zeros

Summary Under regularity conditions the cross-spectrum of a stationary, bivariate, Gaussian process, continuous in mean and with unit variance, may be estimated consistently as T → ∞ from the zeros in the interval [0, T] of its two components and the knowledge that the components have the same or opposite sign at time 0. The power spectra of the components may be estimated from the respective components' zeros alone. The estimates are asymptotically normal and unbiased. During the course of the paper a bound, that may be of independent interest, is given for the joint cumulants of certain functions of multivariate normal random variables. A general theorem on the joint asymptotic normality of spectral and cross-spectral estimates is presented. There is also a worked example.