Local Distributions and Rate Fluctuations in a Unified Scaling Law for Earthquakes

A recently proposed unified scaling law for interoccurrence times of earthquakes [P. Bak et al., Phys. Rev. Lett. {\bf 88}, 178501 (2002)] is analyzed, both theoretically and with data from Southern California. We decompose the corresponding probability density into local-instantaneous distributions, which scale with the rate of earthquake occurrence. The fluctuations of the rate, characterizing the non-stationarity of the process, show a doubly power-law distribution and are fundamental to determine the overall behavior, described by a double power law as well.