Recurrence of extreme events with power‐law interarrival times

Abstract

In various geophysical applications, power‐law interarrival times are observed between extreme events. Classical extreme value theory is based on exponentially distributed interarrivals and can not be applied to these processes. We solve for the density of the maxima of a sequence of random extreme events with any distribution of random interarrivals by applying a continuous time random max model, similar to a random walk model. The equation is exact when the distributions of the exceedances and the interarrivals are known. If only the tail properties of the exceedances and interarrivals can be estimated, then limiting extreme value distributions governing the maximum observation or exceedance are used. The general extreme value densities are obtained by transforming the classical extreme value distributions via subordination. This new class of extreme value densities can be used to obtain recurrence intervals for extreme events with power‐law interarrivals.