Discrete scale-invariance in cross-correlations between time series

•A new method is proposed to detect discrete long-range correlations.•Sectors in a same stock turn out to have continuous long-range correlations.•Sectors in different stock markets prove to have discrete scaling invariance.

The de-trended cross-correlation analysis (DCCA) is converted to a new form, which turns out to be a periodic function modulated power-law, to evaluate discrete-scale long-range cross-correlation between time series. If the modulator is dominated with one frequency, the derived form will degenerate to a log-periodic power-law. We investigate a total of five important stock markets distributing in different continents. Calculations show that the cross-correlations between different stock markets may hint at log-periodic oscillations. This finding may be helpful for us to evaluate financial state in a global way.