On rare and extreme events

In many fields of applied physics, the space-time phenomena to be studied may be described in the following way: events of random amplitudes occur randomly in time. We investigate some statistical properties of this model, with special emphasis on situations where the model for the waiting time between consecutive events or the amplitude of individual events are fractal (power-law) distributions with infinite mean value (the rareness or extreme event hypothesis). Limit laws for cumulative partial sums and the extremal process are characterized. Using asymptotical results on backward and forward recurrence times, limit laws are investigated for the physically realistic situation when the cumulative process is only observed starting from some non-zero observational time.