Coupled Continuous Time Random Maxima

Abstract: 

Continuous Time Random Maxima (CTRM) are a generalization of the classical Extreme value theory: Instead of observing random events at regular intervals in time or with exponential distributed interarrival times, the waiting times between the events are also arbitrarily random variables. 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. Additionally we assume hat the waiting times are coupled to the preceding events. For analysing this we use harmonic analysis on semigroups which gives us a generalized Laplace transform and a L\'evy-Khintchine formula on the semigroup ([0,∞)×[−∞,+∞][0,∞)×[−∞,+∞],∨+∨+). With a continuous mapping approach we derive a limit theorem for the long-time behaviour of the process. Furthermore we get the distribution function of the limit process and a formula for the Laplace transform of the distribution function of the limit distribution. With this formula we have another way to calculate the distribution function of the limit process, namely by inversion of the Laplace transform. We get also governing equations, which are in our case time fractional differential equations whose solutions are the distribution functions of our limit processes.