Analytical error estimate for the cross-correlation, phase and time lag between two light curves

Temporal analysis of radiation from Astrophysical sources like Active Galactic Nuclei, X-ray Binaries and Gamma-ray bursts provide information on the geometry and sizes of the emitting regions. Robustly establishing that two light-curves in different energy bands are correlated and measuring the phase and time-lag between them is an important and frequently used temporal diagnostic. Analytical expressions to estimate the errors on the cross-correlation, phase and time-lag between two light-curves are presented. Earlier estimates depended upon numerically expensive simulations or on dividing the light-curves in large number of segments to find the variance. Thus, the analytical estimates presented here allow for analysis of light-curves with relatively small (∼ 1000) number of points, as well as to obtain information on the longest time-scales available. The error estimation is verified using simulations of light-curves derived from both white and 1/f stochastic processes with measurement errors. As a demonstration, we apply this technique to the XMM-Newton light-curves of the Active Galactic Nucleus, Akn 564.