Coupled Continuous Time Random Maxima

Abstract

Continuous Time Random Maxima (CTRM) are a generalization of classical extreme value theory: Instead of observing random events at regular intervals in time, the waiting times between the events are also random variables which have arbitrary distributions. In case that the waiting times between the events have infinite mean, the limit process that appears differs from the limit process that appears in the classical case. With a continuous mapping approach, we derive a limit theorem for the case that the waiting times and the subsequent events are dependent as well as for the case that the waiting times depend on the preceding events (in this case we speak of an Overshooting Continuous Time Random Maxima, abbr. OCTRM). We get the distribution functions of the limit processes and a formula for the Laplace transform in time of the CTRM and the OCTRM limit. With this formula we have another way to calculate the distribution functions of the limit processes, namely by inversion of the Laplace transform. Moreover, we present governing equations which in our case are time fractional differential equations whose solutions are the distribution functions of our limit processes.