We present a practical implementation of a Monte Carlo method to estimate the significance of cross-correlations in unevenly sampled time series of data, whose statistical properties are modeled with a simple power-law power spectral density. This im
plementation builds on published methods, we introduce a number of improvements in the normalization of the cross-correlation function estimate and a bootstrap method for estimating the significance of the cross-correlations. A closely related matter is the estimation of a model for the light curves, which is critical for the significance estimates. We present a graphical and quantitative demonstration that uses simulations to show how common it is to get high cross-correlations for unrelated light curves with steep power spectral densities. This demonstration highlights the dangers of interpreting them as signs of a physical connection. We show that by using interpolation and the Hanning sampling window function we are able to reduce the effects of red-noise leakage and to recover steep simple power-law power spectral densities. We also introduce the use of a Neyman construction for the estimation of the errors in the power-law index of the power spectral density. This method provides a consistent way to estimate the significance of cross-correlations in unevenly sampled time series of data.
In order to determine the location of the gamma-ray emission site in blazars, we investigate the time-domain relationship between their radio and gamma-ray emission. Light-curves for the brightest detected blazars from the first 3 years of the missio
n of the Fermi Gamma-ray Space Telescope are cross-correlated with 4 years of 15GHz observations from the OVRO 40-m monitoring program. The large sample and long light-curve duration enable us to carry out a statistically robust analysis of the significance of the cross-correlations, which is investigated using Monte Carlo simulations including the uneven sampling and noise properties of the light-curves. Modeling the light-curves as red noise processes with power-law power spectral densities, we find that only one of 41 sources with high quality data in both bands shows correlations with significance larger than 3-sigma (AO 0235+164), with only two more larger than even 2.25-sigma (PKS 1502+106 and B2 2308+34). Additionally, we find correlated variability in Mrk 421 when including a strong flare that occurred in July-September 2012. These results demonstrate very clearly the difficulty of measuring statistically robust multiwavelength correlations and the care needed when comparing light-curves even when many years of data are used. This should be a caution. In all four sources the radio variations lag the gamma-ray variations, suggesting that the gamma-ray emission originates upstream of the radio emission. Continuous simultaneous monitoring over a longer time period is required to obtain high significance levels in cross-correlations between gamma-ray and radio variability in most blazars.
