On the Probability Distribution and Null-hypothesis Testing of Cross-correlation for Light Curves in Active Galactic Nuclei

Cross-correlation is crucial in studies of multiwavelength flux variability in active galactic nuclei (AGNs), especially for reverberation mapping analysis, where interpolated cross-correlation function is widely used to measure time lags between light curves. While time-lag uncertainties can be estimated via the flux randomization and random subset selection method, an appropriate framework for assessing cross-correlation significance remains lacking in the literature. Here we attempt to fill this gap by leveraging the well-established property from stochastic time series theory, namely that the probability distribution of cross-correlation coefficients for independent stochastic light curves asymptotically approaches a normal distribution. Its variance can be analytically estimated using the auto-correlation functions of the light curves. We employ Monte Carlo simulations to validate this property for irregularly sampled, red-noise AGN light curves, and then propose a fast procedure to perform null-hypothesis testing for the cross-correlation of AGN light curves. We also present exemplary applications to AGN reverberation mapping data. This procedure requires prior determination of the auto-correlation functions of light curves, which can be obtained via model fitting. The long-standing issue regarding unbiasedly recovering the auto-correlation function remains unresolved when light-curve duration is comparable to the typical variation timescale, warranting further future investigation.