  The driving concept behind one of the most successful statistical forecasting
models, the ETAS model, has been that the seismicity is driven by spontaneously
occurring background earthquakes that cascade into multitudes of triggered
earthquakes. In nearly all generalizations of the ETAS model, the magnitudes of
the background and the triggered earthquakes are assumed to follow
Gutenberg-Richter law with the same exponent (\b{eta}-value). Furthermore, the
magnitudes of the triggered earthquakes are always assumed to be independent of
the magnitude of the triggering earthquake. Using an EM algorithm applied to
the Californian earthquake catalogue, we show that the distribution of
earthquake magnitudes exhibits three distinct \b{eta}-values: \b{eta}_b for
background events; \b{eta}_a-{\delta} and \b{eta}_a+{\delta}, respectively, for
triggered events below and above the magnitude of the triggering earthquake;
the two last values express a correlation between the magnitudes of triggered
events with that of the triggering earthquake, a feature so far absent in all
proposed operational generalizations of the ETAS model. The ETAS model
incorporating this kinked magnitude distribution provides by far the best
description of seismic catalogs and could thus have the best forecasting
potential. We speculate that the kinked magnitude distribution may result from
the system tending to restore the symmetry of the regional displacement
gradient tensor that has been broken by the initiating event. The general
emerging concept could be that while the background events occur primarily to
accommodate the symmetric stress tensor at the boundaries of the system, the
triggered earthquakes are quasi-Goldstone fluctuations of a self-organized
critical deformation state.
