Statistical tests for power-law cross-correlated processes

For stationary time series, the cross-covariance and the cross-correlation as functions of time lag n serve to quantify the similarity of two time series. The latter measure is also used to assess whether the cross-correlations are statistically significant. For nonstationary time series, the analogous measures are detrended cross-correlations analysis (DCCA) and the recently proposed detrended cross-correlation coefficient, rho(DCCA)(T,n), where T is the total length of the time series and n the window size. For rho(DCCA)(T,n), we numerically calculated the Cauchy inequality-1 <= rho(DCCA)(T,n) <= 1. Here we derive -1 <= rho(DCCA)(T, n) <= 1 for a standard variance-covariance approach and for a detrending approach. For overlapping windows, we find the range of rho(DCCA) within which the cross-correlations become statistically significant. For overlapping windows we numerically determine-and for nonoverlapping windows we derive-that the standard deviation of rho(DCCA)(T,n) tends with increasing T to 1/T. Using rho(DCCA)(T, n) we show that the Chinese financial market's tendency to follow the U.S. market is extremely weak. We also propose an additional statistical test that can be used to quantify the existence of cross-correlations between two power-law correlated time series.