  This chapter discusses correlation analysis of stationary multivariate
Gaussian time series in the spectral or Fourier domain. The goal is to identify
the hub time series, i.e., those that are highly correlated with a specified
number of other time series. We show that Fourier components of the time series
at different frequencies are asymptotically statistically independent. This
property permits independent correlation analysis at each frequency,
alleviating the computational and statistical challenges of high-dimensional
time series. To detect correlation hubs at each frequency, an existing
correlation screening method is extended to the complex numbers to accommodate
complex-valued Fourier components. We characterize the number of hub
discoveries at specified correlation and degree thresholds in the regime of
increasing dimension and fixed sample size. The theory specifies appropriate
thresholds to apply to sample correlation matrices to detect hubs and also
allows statistical significance to be attributed to hub discoveries. Numerical
results illustrate the accuracy of the theory and the usefulness of the
proposed spectral framework.
