  The concept of memory is of central importance for characterizing complex
systems and phenomena. Presence of long-term memories indicates how their
dynamics can be less sensitive to initial conditions compared to the chaotic
cases. On the other hand, it is empirically known that the Feller-Pareto
distribution, which decays as the power law i.e. the scale-invariant nature,
frequently appears as a statistical law generated by the dynamics of complex
systems. However, it is generally not a simple task to determine if a system
obeying such a power law possesses a high degree of complexity with a long-term
memory. Here, a new method is proposed for characterization of memory. In
particular, a scaling relation to be satisfied by any memoryless dynamics
generating the Feller-Pareto power-law distribution is presented. Then, the
method is applied to the real data of energies released by a series of
earthquakes and acceleration of ground motion due to a strong earthquake. It is
shown in this way that the sequence of the released energy in seismicity is
memoryless in the event time, whereas that of acceleration is memoryful in the
sampling time.
