The statistical properties of the cross spectrum

The cross spectrum encodes the correlated variability between two time signals. In recent years, the cross spectrum has been used to study astronomical sources, particularly in the field of X-ray timing. In the literature, it has been common to either simultaneously fit the real and imaginary components of the cross spectrum, or fit the phase and magnitude. Until now, a full discussion of the statistical distribution of the cross spectrum has been missing from the astronomical literature. In this paper, we present a derivation of the full statistical distribution of a cross spectrum between two time series, showing that it follows an asymmetric Laplace distribution. We further provide the probability distribution function for a cross-spectrum random variable, along with the marginal distributions for many quantities. We also relate the cross spectrum to the power spectra of the constituent time series. This work will enable the cross spectrum to be used more accurately as a probe of physical processes such as accretion onto black holes and neutron stars.