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 with 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 and for the case that the waiting times dependent 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 a Laplace transform for 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 are in our case time fractional differential equations whose solutions are the distribution functions of our limit processes. Because of the inverse relationship between the CTRM and its first hitting time we get also the Laplace transform of the distribution function of the first hitting time.