An Introduction to Polyspectra

The subject of this paper is the higher-order spectra or polyspectra of multivariate stationary time series. The intent is to derive (i) certain mathematical properties of polyspectra, (ii) estimates of polyspectra based on an observed stretch of time series, (iii) certain statistical properties of the proposed estimates and (iv) several applications of the results obtained. As might be expected, in lower order cases the polyspectrum reduces to spectra already considered. If one is considering a single time series, the first order polyspectrum is the usual power spectrum considered in 2, 14, 22, while the second order polyspectrum is the bispectrum considered in 12, 23, 28. Also, if one is considering a pair of time series the first order polyspectrum is the cross-spectrum considered in 6, 10, 15. For the case of a single time series the idea of a higher-order spectrum occurs in 3. The idea has since been developed to a higher level of algebraic and analytic detail in 24. Also in 24 the notion of considering a spectral representation for a cumulant rather than for a product moment occurs and is acknowledged to be due to Kolmogorov. Another related early paper is 18. The present paper generalizes the definitions of these papers in the sense that k-dimensional time series are considered. Another contribution is a theorem indicating that for a broad class of processes one is wise to restrict consideration to cumulants rather than product moments. Finally it should be noted that the term polyspectrum is due to J. W. Tukey. I have perhaps used the term in a more restricted sense than he would wish in that I have reserved it for the Fourier transform of a cumulant (at the expense of other functions of moments).