  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.
