The use of stable laws in seismicity models
A new model for the spatial distribution of seismicity (earthquake rate) is proposed. It is derived from the self-similarity of that distribution. Poisson rates of earthquake occurrence for nonoverlapping areas appear to be distributed in accordance with a law belonging to the family of stable laws. Recurrence relations for the distribution of earthquakes occurring in an area are derived. A statistical technique for parameter estimation of the stable law is presented. The technique is illustrated by processing the earthquake catalogs for Pamir--Tien Shan, Japan, and California. The $\alpha$ parameters of stable laws were found to be approximately equal to 0.6--0.7. The necessity for a correct statistical data processing is emphasized, since for these $\alpha$ values the expectation and variance of the stable law are unbounded.