On rare and extreme events

In many fields of applied physics, the space-time phenomena to be studied m ay be described in the following way: events of random amplitudes occur ran domly in time. We investigate some statistical properties of this model, wi th special emphasis on situations where the model for the waiting time betw een consecutive events or the amplitude of individual events are fractal (p ower-law) distributions with infinite mean value (the rareness or extreme e vent hypothesis). Limit laws for cumulative partial sums and the extremal p rocess are characterized. Using asymptotical results on backward and forwar d recurrence times, limit laws are investigated for the physically realisti c situation when the cumulative process is only observed starting from some non-zero observational time. (C) 2001 Elsevier Science Ltd. All rights res erved.