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 an 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 autocorrelation 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 autocorrelation functions of light curves, which can be obtained via model fitting. The long-standing issue regarding unbiasedly recovering the autocorrelation function remains unresolved when light-curve duration is comparable to the typical variation timescale, warranting further future investigation.